| L(s) = 1 | + 1.17·3-s − 3.43·5-s − 1.61·9-s + 2.61·11-s − 5.43·13-s − 4.04·15-s + 0.611·17-s + 19-s − 1.43·23-s + 6.79·25-s − 5.43·27-s + 1.74·29-s − 0.255·31-s + 3.07·33-s + 5.79·37-s − 6.40·39-s − 8.96·41-s + 7.58·43-s + 5.53·45-s + 10.6·47-s + 0.720·51-s + 5.53·53-s − 8.96·55-s + 1.17·57-s − 4.30·59-s + 1.27·61-s + 18.6·65-s + ⋯ |
| L(s) = 1 | + 0.680·3-s − 1.53·5-s − 0.537·9-s + 0.787·11-s − 1.50·13-s − 1.04·15-s + 0.148·17-s + 0.229·19-s − 0.298·23-s + 1.35·25-s − 1.04·27-s + 0.323·29-s − 0.0458·31-s + 0.535·33-s + 0.951·37-s − 1.02·39-s − 1.40·41-s + 1.15·43-s + 0.825·45-s + 1.55·47-s + 0.100·51-s + 0.760·53-s − 1.20·55-s + 0.156·57-s − 0.559·59-s + 0.163·61-s + 2.31·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.286377602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.286377602\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.17T + 3T^{2} \) |
| 5 | \( 1 + 3.43T + 5T^{2} \) |
| 11 | \( 1 - 2.61T + 11T^{2} \) |
| 13 | \( 1 + 5.43T + 13T^{2} \) |
| 17 | \( 1 - 0.611T + 17T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 - 1.74T + 29T^{2} \) |
| 31 | \( 1 + 0.255T + 31T^{2} \) |
| 37 | \( 1 - 5.79T + 37T^{2} \) |
| 41 | \( 1 + 8.96T + 41T^{2} \) |
| 43 | \( 1 - 7.58T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 5.53T + 53T^{2} \) |
| 59 | \( 1 + 4.30T + 59T^{2} \) |
| 61 | \( 1 - 1.27T + 61T^{2} \) |
| 67 | \( 1 - 0.611T + 67T^{2} \) |
| 71 | \( 1 + 1.07T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 7.88T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.477056226897602873648869876078, −7.67152825678618035929650073335, −7.42228490380432989464561480579, −6.47041209633109383001378352288, −5.40207560207854206368336541064, −4.48928696800313816782977062905, −3.84027038882360792931192399519, −3.07700190643994908161055391844, −2.23442062372540720502705577906, −0.61449861086728187076872324367,
0.61449861086728187076872324367, 2.23442062372540720502705577906, 3.07700190643994908161055391844, 3.84027038882360792931192399519, 4.48928696800313816782977062905, 5.40207560207854206368336541064, 6.47041209633109383001378352288, 7.42228490380432989464561480579, 7.67152825678618035929650073335, 8.477056226897602873648869876078
