| L(s) = 1 | − 2.70·2-s + 3-s + 5.34·4-s − 2.70·6-s + 7-s − 9.04·8-s + 9-s + 2·11-s + 5.34·12-s + 0.921·13-s − 2.70·14-s + 13.8·16-s + 1.07·17-s − 2.70·18-s + 3.07·19-s + 21-s − 5.41·22-s + 2.34·23-s − 9.04·24-s − 2.49·26-s + 27-s + 5.34·28-s − 6.68·29-s − 7.75·31-s − 19.3·32-s + 2·33-s − 2.92·34-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 0.577·3-s + 2.67·4-s − 1.10·6-s + 0.377·7-s − 3.19·8-s + 0.333·9-s + 0.603·11-s + 1.54·12-s + 0.255·13-s − 0.724·14-s + 3.45·16-s + 0.261·17-s − 0.638·18-s + 0.706·19-s + 0.218·21-s − 1.15·22-s + 0.487·23-s − 1.84·24-s − 0.489·26-s + 0.192·27-s + 1.00·28-s − 1.24·29-s − 1.39·31-s − 3.42·32-s + 0.348·33-s − 0.501·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8404594786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8404594786\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 0.921T + 13T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 - 3.07T + 19T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 7.75T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 + 6.52T + 43T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 4.15T + 61T^{2} \) |
| 67 | \( 1 - 4.68T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 - 6.15T + 79T^{2} \) |
| 83 | \( 1 - 6.83T + 83T^{2} \) |
| 89 | \( 1 - 8.34T + 89T^{2} \) |
| 97 | \( 1 + 8.43T + 97T^{2} \) |
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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
−10.73093724426907748138347685787, −9.488955697269998902726127343570, −9.311266427577489277333430127999, −8.260314135259506262863829014208, −7.59127026891978329060420902585, −6.82472178482904034539484641598, −5.63385722833276316572021417302, −3.66481207120444030082643463708, −2.33657418047064760612703806245, −1.14951268463002916512455010277,
1.14951268463002916512455010277, 2.33657418047064760612703806245, 3.66481207120444030082643463708, 5.63385722833276316572021417302, 6.82472178482904034539484641598, 7.59127026891978329060420902585, 8.260314135259506262863829014208, 9.311266427577489277333430127999, 9.488955697269998902726127343570, 10.73093724426907748138347685787
