| L(s) = 1 | − 3.53·2-s + 4.46·4-s + 7·7-s + 12.4·8-s + 2.93·11-s + 19.0·13-s − 24.7·14-s − 79.7·16-s + 122.·17-s + 107.·19-s − 10.3·22-s + 210.·23-s − 67.3·26-s + 31.2·28-s − 95.4·29-s − 94.3·31-s + 181.·32-s − 432.·34-s − 97.1·37-s − 379.·38-s + 491.·41-s + 43.0·43-s + 13.1·44-s − 743.·46-s + 473.·47-s + 49·49-s + 85.1·52-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.558·4-s + 0.377·7-s + 0.551·8-s + 0.0805·11-s + 0.406·13-s − 0.471·14-s − 1.24·16-s + 1.74·17-s + 1.29·19-s − 0.100·22-s + 1.90·23-s − 0.507·26-s + 0.211·28-s − 0.611·29-s − 0.546·31-s + 1.00·32-s − 2.18·34-s − 0.431·37-s − 1.61·38-s + 1.87·41-s + 0.152·43-s + 0.0449·44-s − 2.38·46-s + 1.46·47-s + 0.142·49-s + 0.227·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.437857769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.437857769\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 3.53T + 8T^{2} \) |
| 11 | \( 1 - 2.93T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 210.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 95.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 97.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 491.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 43.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 473.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 183.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 760.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 198.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 309.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 665.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 621.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 24.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 406.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 261.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.00e3T + 9.12e5T^{2} \) |
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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
−9.153353705586358740942453497971, −8.368641941146238382480236027923, −7.43914919027476010469579570411, −7.24818549978646837826075463789, −5.76931935054066266328814399972, −5.08914932039572729156213172901, −3.88725987228588153584368358120, −2.81147344313458567046704702605, −1.37364563684643671998184723910, −0.825756224556272554430617877710,
0.825756224556272554430617877710, 1.37364563684643671998184723910, 2.81147344313458567046704702605, 3.88725987228588153584368358120, 5.08914932039572729156213172901, 5.76931935054066266328814399972, 7.24818549978646837826075463789, 7.43914919027476010469579570411, 8.368641941146238382480236027923, 9.153353705586358740942453497971
