| L(s) = 1 | + 3.22·3-s + 1.65i·7-s + 7.41·9-s + 4.44i·11-s + (2.97 + 2.04i)13-s − 5.33·17-s + 0.472i·19-s + 5.33i·21-s − 1.08·23-s + 14.2·27-s − 5.50·29-s + 5.47i·31-s + 14.3i·33-s + 3.65i·37-s + (9.59 + 6.58i)39-s + ⋯ |
| L(s) = 1 | + 1.86·3-s + 0.625i·7-s + 2.47·9-s + 1.34i·11-s + (0.824 + 0.566i)13-s − 1.29·17-s + 0.108i·19-s + 1.16i·21-s − 0.226·23-s + 2.73·27-s − 1.02·29-s + 0.982i·31-s + 2.49i·33-s + 0.600i·37-s + (1.53 + 1.05i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.647943149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.647943149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-2.97 - 2.04i)T \) |
| good | 3 | \( 1 - 3.22T + 3T^{2} \) |
| 7 | \( 1 - 1.65iT - 7T^{2} \) |
| 11 | \( 1 - 4.44iT - 11T^{2} \) |
| 17 | \( 1 + 5.33T + 17T^{2} \) |
| 19 | \( 1 - 0.472iT - 19T^{2} \) |
| 23 | \( 1 + 1.08T + 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 - 5.47iT - 31T^{2} \) |
| 37 | \( 1 - 3.65iT - 37T^{2} \) |
| 41 | \( 1 + 10.4iT - 41T^{2} \) |
| 43 | \( 1 - 8.14T + 43T^{2} \) |
| 47 | \( 1 + 1.65iT - 47T^{2} \) |
| 53 | \( 1 + 4.14T + 53T^{2} \) |
| 59 | \( 1 - 2.52iT - 59T^{2} \) |
| 61 | \( 1 - 0.160T + 61T^{2} \) |
| 67 | \( 1 - 8.26iT - 67T^{2} \) |
| 71 | \( 1 + 12.9iT - 71T^{2} \) |
| 73 | \( 1 + 10.2iT - 73T^{2} \) |
| 79 | \( 1 + 8.65T + 79T^{2} \) |
| 83 | \( 1 + 14.0iT - 83T^{2} \) |
| 89 | \( 1 - 7.61iT - 89T^{2} \) |
| 97 | \( 1 + 16.7iT - 97T^{2} \) |
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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
−8.864630578476247571617954419195, −8.559050184675564210882326583730, −7.43022528014981289012115920204, −7.10168267019448967942293587728, −6.05334748711448665383507324276, −4.69919460389394493222247074389, −4.10597305803721693061867303141, −3.23004595915350682747225259684, −2.16109106444297900421409103862, −1.78893856720956338228675344294,
0.950782544221473328299216402275, 2.18974928210408963695823158547, 3.04671880008335309359555932698, 3.80431669689268855038315682810, 4.34970052427376633364869511863, 5.76504436768863972554372230652, 6.63926801697882489102602788264, 7.56139239434245827346451824401, 8.084219147984023337793633535352, 8.730062003508166046962727617055
