| L(s) = 1 | − 3-s − 3.23·5-s + 2.16·7-s + 9-s − 1.14·11-s − 2.10·13-s + 3.23·15-s + 7.69·17-s + 6.58·19-s − 2.16·21-s + 7.25·23-s + 5.48·25-s − 27-s − 3.25·29-s − 6.11·31-s + 1.14·33-s − 7.02·35-s − 5.01·37-s + 2.10·39-s − 4.15·41-s − 6.96·43-s − 3.23·45-s + 8.84·47-s − 2.29·49-s − 7.69·51-s + 4.17·53-s + 3.71·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.44·5-s + 0.819·7-s + 0.333·9-s − 0.345·11-s − 0.584·13-s + 0.835·15-s + 1.86·17-s + 1.51·19-s − 0.473·21-s + 1.51·23-s + 1.09·25-s − 0.192·27-s − 0.603·29-s − 1.09·31-s + 0.199·33-s − 1.18·35-s − 0.825·37-s + 0.337·39-s − 0.649·41-s − 1.06·43-s − 0.482·45-s + 1.29·47-s − 0.327·49-s − 1.07·51-s + 0.574·53-s + 0.500·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.232853850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.232853850\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 2.16T + 7T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 13 | \( 1 + 2.10T + 13T^{2} \) |
| 17 | \( 1 - 7.69T + 17T^{2} \) |
| 19 | \( 1 - 6.58T + 19T^{2} \) |
| 23 | \( 1 - 7.25T + 23T^{2} \) |
| 29 | \( 1 + 3.25T + 29T^{2} \) |
| 31 | \( 1 + 6.11T + 31T^{2} \) |
| 37 | \( 1 + 5.01T + 37T^{2} \) |
| 41 | \( 1 + 4.15T + 41T^{2} \) |
| 43 | \( 1 + 6.96T + 43T^{2} \) |
| 47 | \( 1 - 8.84T + 47T^{2} \) |
| 53 | \( 1 - 4.17T + 53T^{2} \) |
| 59 | \( 1 - 1.07T + 59T^{2} \) |
| 61 | \( 1 + 2.11T + 61T^{2} \) |
| 67 | \( 1 + 8.68T + 67T^{2} \) |
| 71 | \( 1 - 8.82T + 71T^{2} \) |
| 73 | \( 1 - 0.732T + 73T^{2} \) |
| 79 | \( 1 + 1.27T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 7.36T + 89T^{2} \) |
| 97 | \( 1 + 4.11T + 97T^{2} \) |
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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
−7.87287804074744886995761127332, −7.31723741280002886478262320374, −7.10905619802289706510206206659, −5.58257698913403504471058181020, −5.25950627593568760385640982129, −4.58695510827949707676685862400, −3.55970042522748563895699990675, −3.10839673007680739312655662526, −1.57292243347609878957056739816, −0.63667079771485723661844942523,
0.63667079771485723661844942523, 1.57292243347609878957056739816, 3.10839673007680739312655662526, 3.55970042522748563895699990675, 4.58695510827949707676685862400, 5.25950627593568760385640982129, 5.58257698913403504471058181020, 7.10905619802289706510206206659, 7.31723741280002886478262320374, 7.87287804074744886995761127332
