| L(s) = 1 | + 2·9-s − 8·13-s − 12·17-s + 24·25-s − 12·29-s + 12·37-s − 36·41-s − 6·49-s + 48·53-s − 12·61-s + 7·81-s − 36·89-s + 12·97-s + 36·101-s + 36·113-s − 16·117-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 2.21·13-s − 2.91·17-s + 24/5·25-s − 2.22·29-s + 1.97·37-s − 5.62·41-s − 6/7·49-s + 6.59·53-s − 1.53·61-s + 7/9·81-s − 3.81·89-s + 1.21·97-s + 3.58·101-s + 3.38·113-s − 1.47·117-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6355750485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6355750485\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| good | 3 | \( 1 - 2 T^{2} - p T^{4} + 22 T^{6} - 68 T^{8} + 22 p^{2} T^{10} - p^{5} T^{12} - 2 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 7 | \( 1 + 6 T^{2} + 37 T^{4} - 594 T^{6} - 4164 T^{8} - 594 p^{2} T^{10} + 37 p^{4} T^{12} + 6 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 + 14 T^{2} + 13 T^{4} - 826 T^{6} - 4868 T^{8} - 826 p^{2} T^{10} + 13 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 6 T + 5 T^{2} - 18 T^{3} + 60 T^{4} - 18 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 54 T^{2} + 1573 T^{4} + 33534 T^{6} + 620652 T^{8} + 33534 p^{2} T^{10} + 1573 p^{4} T^{12} + 54 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 10 T^{2} - 875 T^{4} + 830 T^{6} + 617884 T^{8} + 830 p^{2} T^{10} - 875 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 36 T^{2} + 518 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2 T + p T^{2} )^{4}( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | \( ( 1 + 18 T + 213 T^{2} + 1890 T^{3} + 13772 T^{4} + 1890 p T^{5} + 213 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 82 T^{2} + 2317 T^{4} - 58138 T^{6} + 3570172 T^{8} - 58138 p^{2} T^{10} + 2317 p^{4} T^{12} - 82 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 4 T^{2} + 2694 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 - 10 T^{2} - 5915 T^{4} + 9470 T^{6} + 23714764 T^{8} + 9470 p^{2} T^{10} - 5915 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 246 T^{2} + 36517 T^{4} + 3695166 T^{6} + 284144556 T^{8} + 3695166 p^{2} T^{10} + 36517 p^{4} T^{12} + 246 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 + 14 T^{2} - 1187 T^{4} - 121786 T^{6} - 24487028 T^{8} - 121786 p^{2} T^{10} - 1187 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 16 T + p T^{2} )^{4}( 1 + 16 T + p T^{2} )^{4} \) |
| 79 | \( ( 1 + 196 T^{2} + 20358 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 212 T^{2} + 23286 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 18 T + 297 T^{2} + 3402 T^{3} + 37412 T^{4} + 3402 p T^{5} + 297 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 6 T + 65 T^{2} - 318 T^{3} - 5436 T^{4} - 318 p T^{5} + 65 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.03567224377668976549885540255, −4.95563858632604233034186623596, −4.69965266893849253746246853253, −4.41653676313366192729591143277, −4.40048269377744647480287752684, −4.38956819989534543368259184009, −4.25286770059819553037975205816, −3.89942081095920061023309736248, −3.87742847627620100571773913165, −3.57978343572820033313778666103, −3.50126727094041095816937563672, −3.16735763112841146499750054746, −3.07798037057002553839358610575, −3.00436830595302322900279567023, −2.88254512630175736678088960492, −2.46052767009696544430867724731, −2.39144556708733591034295391988, −2.19619225988201727065988138706, −2.13459194825178128344318409120, −1.85435352348229881135162626374, −1.79651805371411852427655471626, −1.25549760220754058442435390340, −1.05325020756801103424235098021, −0.75640657320093938706198899050, −0.17418240390647659509826736665,
0.17418240390647659509826736665, 0.75640657320093938706198899050, 1.05325020756801103424235098021, 1.25549760220754058442435390340, 1.79651805371411852427655471626, 1.85435352348229881135162626374, 2.13459194825178128344318409120, 2.19619225988201727065988138706, 2.39144556708733591034295391988, 2.46052767009696544430867724731, 2.88254512630175736678088960492, 3.00436830595302322900279567023, 3.07798037057002553839358610575, 3.16735763112841146499750054746, 3.50126727094041095816937563672, 3.57978343572820033313778666103, 3.87742847627620100571773913165, 3.89942081095920061023309736248, 4.25286770059819553037975205816, 4.38956819989534543368259184009, 4.40048269377744647480287752684, 4.41653676313366192729591143277, 4.69965266893849253746246853253, 4.95563858632604233034186623596, 5.03567224377668976549885540255
Plot not available for L-functions of degree greater than 10.
