| L(s) = 1 | + 8·5-s − 40·11-s + 24·13-s + 58·17-s + 26·19-s − 64·23-s + 125·25-s + 124·29-s + 252·31-s − 26·37-s + 12·41-s + 832·43-s + 396·47-s − 450·53-s − 320·55-s − 274·59-s − 576·61-s + 192·65-s + 476·67-s + 896·71-s − 158·73-s + 936·79-s + 1.06e3·83-s + 464·85-s + 390·89-s + 208·95-s − 428·97-s + ⋯ |
| L(s) = 1 | + 0.715·5-s − 1.09·11-s + 0.512·13-s + 0.827·17-s + 0.313·19-s − 0.580·23-s + 25-s + 0.794·29-s + 1.46·31-s − 0.115·37-s + 0.0457·41-s + 2.95·43-s + 1.22·47-s − 1.16·53-s − 0.784·55-s − 0.604·59-s − 1.20·61-s + 0.366·65-s + 0.867·67-s + 1.49·71-s − 0.253·73-s + 1.33·79-s + 1.40·83-s + 0.592·85-s + 0.464·89-s + 0.224·95-s − 0.448·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.260598442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.260598442\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T - 61 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 40 T + 269 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 12 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 58 T - 1549 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 26 T - 6183 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 64 T - 8071 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 62 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 252 T + 33713 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T - 49977 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 416 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 396 T + 52993 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 450 T + 53623 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 274 T - 130303 T^{2} + 274 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 576 T + 104795 T^{2} + 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 476 T - 74187 T^{2} - 476 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 448 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 158 T - 364053 T^{2} + 158 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 936 T + 383057 T^{2} - 936 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 530 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 390 T - 552869 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 214 T + p^{3} T^{2} )^{2} \) |
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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
−9.255887300162183561842786596843, −8.729292828835012802208361705323, −8.245086065824642302060134857514, −8.030667208989621816205845168612, −7.54678592215186116813815639788, −7.31933925779635315391651470283, −6.59129365656045037599730900382, −6.28301496224030883602872981463, −5.93928616435496621750614463877, −5.57401600566506854255533556767, −4.91904295118872928988428800584, −4.89076362681420539501449243390, −4.11149496764637297387107941249, −3.74462660263562090803737911397, −2.87425079118821343491006006516, −2.84714638784033858780352337816, −2.23767287548895698754129496240, −1.59439758247897233860412866604, −0.830480130095331972073021294817, −0.64692539553204180651752249668,
0.64692539553204180651752249668, 0.830480130095331972073021294817, 1.59439758247897233860412866604, 2.23767287548895698754129496240, 2.84714638784033858780352337816, 2.87425079118821343491006006516, 3.74462660263562090803737911397, 4.11149496764637297387107941249, 4.89076362681420539501449243390, 4.91904295118872928988428800584, 5.57401600566506854255533556767, 5.93928616435496621750614463877, 6.28301496224030883602872981463, 6.59129365656045037599730900382, 7.31933925779635315391651470283, 7.54678592215186116813815639788, 8.030667208989621816205845168612, 8.245086065824642302060134857514, 8.729292828835012802208361705323, 9.255887300162183561842786596843
