| L(s) = 1 | − 6·9-s − 32·13-s + 48·17-s + 10·25-s − 48·29-s + 80·37-s − 120·41-s + 52·49-s + 96·53-s + 56·61-s − 8·73-s + 27·81-s + 24·89-s − 56·97-s − 48·101-s + 280·109-s + 576·113-s + 192·117-s + 340·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 288·153-s + 157-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 2.46·13-s + 2.82·17-s + 2/5·25-s − 1.65·29-s + 2.16·37-s − 2.92·41-s + 1.06·49-s + 1.81·53-s + 0.918·61-s − 0.109·73-s + 1/3·81-s + 0.269·89-s − 0.577·97-s − 0.475·101-s + 2.56·109-s + 5.09·113-s + 1.64·117-s + 2.80·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 1.88·153-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.369716174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.369716174\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 2598 T^{4} - 52 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 340 T^{2} + 55302 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 16 T + 222 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 24 T + 542 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 868 T^{2} + 402918 T^{4} - 868 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 377862 T^{4} - 100 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 24 T + 1646 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 964 T^{2} + 927366 T^{4} - 964 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 40 T + 1518 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{4} \) |
| 43 | $D_4\times C_2$ | \( 1 - 5380 T^{2} + 13889382 T^{4} - 5380 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2212 T^{2} + 8033478 T^{4} - 2212 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 48 T + 4574 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 620 T^{2} + 23982342 T^{4} + 620 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 28 T + 1158 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 188 T^{2} - 19408602 T^{4} + 188 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 8060 T^{2} + 10662 p^{2} T^{4} + 8060 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 9942 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12868 T^{2} + 113720838 T^{4} - 12868 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 19780 T^{2} + 177798822 T^{4} - 19780 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 15158 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.644515978466745917765062884235, −8.415162715502546197641717428944, −8.003355003461797680419747308944, −7.69382418144301866457884565608, −7.57458445716484285118464006591, −7.39681379910880963196435175438, −7.10882758471019661522553600375, −6.86598106796148264560872493952, −6.53511652046397327337993089806, −5.90058748072078028904481110090, −5.85049570954715614686803995000, −5.70426983897486499443157401130, −5.11867077292626081941876076488, −5.11662393693540468556568635754, −4.92384509019044727649073198166, −4.24343283940517143716101406408, −4.13303555463671937849870148844, −3.53208496918929497047413446254, −3.25127019077945428682781556632, −3.01732619337867202686063809411, −2.60834843336393715955041397678, −1.95610550847262223300549683189, −1.93185878518049828233458046547, −0.859092968332865621591703998275, −0.52142987771127098182874881750,
0.52142987771127098182874881750, 0.859092968332865621591703998275, 1.93185878518049828233458046547, 1.95610550847262223300549683189, 2.60834843336393715955041397678, 3.01732619337867202686063809411, 3.25127019077945428682781556632, 3.53208496918929497047413446254, 4.13303555463671937849870148844, 4.24343283940517143716101406408, 4.92384509019044727649073198166, 5.11662393693540468556568635754, 5.11867077292626081941876076488, 5.70426983897486499443157401130, 5.85049570954715614686803995000, 5.90058748072078028904481110090, 6.53511652046397327337993089806, 6.86598106796148264560872493952, 7.10882758471019661522553600375, 7.39681379910880963196435175438, 7.57458445716484285118464006591, 7.69382418144301866457884565608, 8.003355003461797680419747308944, 8.415162715502546197641717428944, 8.644515978466745917765062884235
