| L(s) = 1 | + 4·5-s + 20·7-s − 60·11-s − 26·13-s + 84·17-s + 60·19-s − 72·23-s − 210·25-s + 124·29-s + 108·31-s + 80·35-s + 36·37-s + 52·41-s + 32·43-s − 428·47-s − 134·49-s − 380·53-s − 240·55-s − 1.42e3·59-s + 1.01e3·61-s − 104·65-s + 844·67-s − 868·71-s − 60·73-s − 1.20e3·77-s − 272·79-s − 1.25e3·83-s + ⋯ |
| L(s) = 1 | + 0.357·5-s + 1.07·7-s − 1.64·11-s − 0.554·13-s + 1.19·17-s + 0.724·19-s − 0.652·23-s − 1.67·25-s + 0.794·29-s + 0.625·31-s + 0.386·35-s + 0.159·37-s + 0.198·41-s + 0.113·43-s − 1.32·47-s − 0.390·49-s − 0.984·53-s − 0.588·55-s − 3.13·59-s + 2.12·61-s − 0.198·65-s + 1.53·67-s − 1.45·71-s − 0.0961·73-s − 1.77·77-s − 0.387·79-s − 1.65·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 226 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 20 T + 534 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 60 T + 3114 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 84 T + 10582 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 60 T + 6526 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 72 T + 14430 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 124 T - 1586 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 108 T - 16154 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 36 T + 42382 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 52 T + 76666 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 32 T + 150198 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 428 T + 116242 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 380 T + 258142 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1420 T + 882490 T^{2} + 1420 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1012 T + 708206 T^{2} - 1012 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 844 T + 778238 T^{2} - 844 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 868 T + 888050 T^{2} + 868 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 60 T + 114886 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 272 T + 993374 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1252 T + 1118698 T^{2} + 1252 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 572 T + 853306 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 708 T + 1762390 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.322265584324752422589185198668, −8.164890316442592924970392597251, −8.030058552080768642537252782994, −7.62923741148788425101860191527, −7.28710042746846389443493995907, −6.71003736748643046145230469162, −6.16397913383407530949936006017, −5.78838368201344413807248292272, −5.33610701143664500506918011395, −5.17585544328762991393188149535, −4.58520431917712232589822190787, −4.33917167591612613888005850069, −3.54869564968335738168489792091, −3.11533308893695196921795150548, −2.57113665196715262395150298148, −2.23082975137771960912342693578, −1.39323565430890839190788559869, −1.29144325965472644963121988852, 0, 0,
1.29144325965472644963121988852, 1.39323565430890839190788559869, 2.23082975137771960912342693578, 2.57113665196715262395150298148, 3.11533308893695196921795150548, 3.54869564968335738168489792091, 4.33917167591612613888005850069, 4.58520431917712232589822190787, 5.17585544328762991393188149535, 5.33610701143664500506918011395, 5.78838368201344413807248292272, 6.16397913383407530949936006017, 6.71003736748643046145230469162, 7.28710042746846389443493995907, 7.62923741148788425101860191527, 8.030058552080768642537252782994, 8.164890316442592924970392597251, 8.322265584324752422589185198668
