| L(s) = 1 | − 1.80·2-s + 3-s + 1.24·4-s + 5-s − 1.80·6-s − 4.47·7-s + 1.36·8-s + 9-s − 1.80·10-s − 4.73·11-s + 1.24·12-s − 5.26·13-s + 8.06·14-s + 15-s − 4.94·16-s + 0.285·17-s − 1.80·18-s − 6.59·19-s + 1.24·20-s − 4.47·21-s + 8.51·22-s + 6.54·23-s + 1.36·24-s + 25-s + 9.48·26-s + 27-s − 5.56·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.577·3-s + 0.620·4-s + 0.447·5-s − 0.735·6-s − 1.69·7-s + 0.482·8-s + 0.333·9-s − 0.569·10-s − 1.42·11-s + 0.358·12-s − 1.46·13-s + 2.15·14-s + 0.258·15-s − 1.23·16-s + 0.0691·17-s − 0.424·18-s − 1.51·19-s + 0.277·20-s − 0.977·21-s + 1.81·22-s + 1.36·23-s + 0.278·24-s + 0.200·25-s + 1.86·26-s + 0.192·27-s − 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3212595139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3212595139\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 + 5.26T + 13T^{2} \) |
| 17 | \( 1 - 0.285T + 17T^{2} \) |
| 19 | \( 1 + 6.59T + 19T^{2} \) |
| 23 | \( 1 - 6.54T + 23T^{2} \) |
| 29 | \( 1 - 8.62T + 29T^{2} \) |
| 31 | \( 1 + 3.91T + 31T^{2} \) |
| 37 | \( 1 + 5.33T + 37T^{2} \) |
| 41 | \( 1 + 2.06T + 41T^{2} \) |
| 43 | \( 1 + 0.493T + 43T^{2} \) |
| 47 | \( 1 + 7.36T + 47T^{2} \) |
| 53 | \( 1 + 7.29T + 53T^{2} \) |
| 59 | \( 1 + 7.64T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 9.38T + 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 - 7.72T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.118477674346751601843426046326, −7.57313434794929813552907989169, −6.80604812356017641686695112344, −6.38036567294533371872951125613, −5.11728251288551467948136433935, −4.56162805813045365830676260274, −3.15447727906507383419299921729, −2.70678856126865699594982227996, −1.84593464476799426142533064067, −0.32957437499220123982528036563,
0.32957437499220123982528036563, 1.84593464476799426142533064067, 2.70678856126865699594982227996, 3.15447727906507383419299921729, 4.56162805813045365830676260274, 5.11728251288551467948136433935, 6.38036567294533371872951125613, 6.80604812356017641686695112344, 7.57313434794929813552907989169, 8.118477674346751601843426046326
