| L(s) = 1 | + 2-s + 2.97·3-s + 4-s + 4.15·5-s + 2.97·6-s − 7-s + 8-s + 5.83·9-s + 4.15·10-s − 4.80·11-s + 2.97·12-s + 6.61·13-s − 14-s + 12.3·15-s + 16-s + 3.58·17-s + 5.83·18-s − 3.93·19-s + 4.15·20-s − 2.97·21-s − 4.80·22-s − 8.59·23-s + 2.97·24-s + 12.2·25-s + 6.61·26-s + 8.42·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.71·3-s + 0.5·4-s + 1.85·5-s + 1.21·6-s − 0.377·7-s + 0.353·8-s + 1.94·9-s + 1.31·10-s − 1.44·11-s + 0.858·12-s + 1.83·13-s − 0.267·14-s + 3.18·15-s + 0.250·16-s + 0.869·17-s + 1.37·18-s − 0.902·19-s + 0.928·20-s − 0.648·21-s − 1.02·22-s − 1.79·23-s + 0.606·24-s + 2.45·25-s + 1.29·26-s + 1.62·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.307399961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.307399961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 2.97T + 3T^{2} \) |
| 5 | \( 1 - 4.15T + 5T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 13 | \( 1 - 6.61T + 13T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 19 | \( 1 + 3.93T + 19T^{2} \) |
| 23 | \( 1 + 8.59T + 23T^{2} \) |
| 29 | \( 1 + 7.98T + 29T^{2} \) |
| 31 | \( 1 + 3.25T + 31T^{2} \) |
| 37 | \( 1 - 1.37T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 4.29T + 43T^{2} \) |
| 47 | \( 1 + 2.41T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 1.93T + 59T^{2} \) |
| 61 | \( 1 - 2.36T + 61T^{2} \) |
| 67 | \( 1 + 3.21T + 67T^{2} \) |
| 71 | \( 1 - 6.61T + 71T^{2} \) |
| 73 | \( 1 + 4.14T + 73T^{2} \) |
| 79 | \( 1 + 4.53T + 79T^{2} \) |
| 83 | \( 1 - 5.57T + 83T^{2} \) |
| 89 | \( 1 + 1.38T + 89T^{2} \) |
| 97 | \( 1 + 0.909T + 97T^{2} \) |
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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
−8.012714508396987710957172775441, −7.57899274472814551051080105264, −6.39269944160399958250292217961, −5.95019590527254008267170829869, −5.37373010633772421492130579015, −4.17029641304971619964208289613, −3.51005631130891521877914635460, −2.71997209353657967844339193373, −2.12621695637665694008697408080, −1.50090426620901830428947793783,
1.50090426620901830428947793783, 2.12621695637665694008697408080, 2.71997209353657967844339193373, 3.51005631130891521877914635460, 4.17029641304971619964208289613, 5.37373010633772421492130579015, 5.95019590527254008267170829869, 6.39269944160399958250292217961, 7.57899274472814551051080105264, 8.012714508396987710957172775441
