L(s) = 1 | + 2.44·3-s + 5-s + 4.02·7-s + 2.98·9-s − 1.17·11-s + 5.60·13-s + 2.44·15-s − 3.00·17-s − 1.62·19-s + 9.85·21-s + 0.296·23-s + 25-s − 0.0323·27-s + 0.296·29-s + 1.77·31-s − 2.88·33-s + 4.02·35-s − 37-s + 13.7·39-s − 1.17·41-s + 5.07·43-s + 2.98·45-s + 4.94·47-s + 9.23·49-s − 7.35·51-s − 8.75·53-s − 1.17·55-s + ⋯ |
L(s) = 1 | + 1.41·3-s + 0.447·5-s + 1.52·7-s + 0.995·9-s − 0.355·11-s + 1.55·13-s + 0.631·15-s − 0.728·17-s − 0.372·19-s + 2.15·21-s + 0.0617·23-s + 0.200·25-s − 0.00623·27-s + 0.0549·29-s + 0.318·31-s − 0.501·33-s + 0.681·35-s − 0.164·37-s + 2.19·39-s − 0.183·41-s + 0.774·43-s + 0.445·45-s + 0.721·47-s + 1.31·49-s − 1.02·51-s − 1.20·53-s − 0.158·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.131230755\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.131230755\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 7 | \( 1 - 4.02T + 7T^{2} \) |
| 11 | \( 1 + 1.17T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 + 3.00T + 17T^{2} \) |
| 19 | \( 1 + 1.62T + 19T^{2} \) |
| 23 | \( 1 - 0.296T + 23T^{2} \) |
| 29 | \( 1 - 0.296T + 29T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 - 5.07T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 + 8.75T + 53T^{2} \) |
| 59 | \( 1 + 6.72T + 59T^{2} \) |
| 61 | \( 1 + 8.22T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 2.74T + 73T^{2} \) |
| 79 | \( 1 + 6.18T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 5.13T + 89T^{2} \) |
| 97 | \( 1 + 3.32T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.793440736168407730197108417321, −8.080543125661124067775237308987, −7.62418053005082487591188126299, −6.52831131136388305854701277052, −5.66894456904063918405783991383, −4.66911445639974772699884828000, −3.97982514499059464350540393590, −2.96671525834612210855475185995, −2.07389743872694472129760557503, −1.36700253444496342444058046329,
1.36700253444496342444058046329, 2.07389743872694472129760557503, 2.96671525834612210855475185995, 3.97982514499059464350540393590, 4.66911445639974772699884828000, 5.66894456904063918405783991383, 6.52831131136388305854701277052, 7.62418053005082487591188126299, 8.080543125661124067775237308987, 8.793440736168407730197108417321