| L(s) = 1 | − 0.567·2-s − 3-s − 1.67·4-s + 5-s + 0.567·6-s + 3.22·7-s + 2.08·8-s + 9-s − 0.567·10-s + 0.237·11-s + 1.67·12-s − 1.83·13-s − 1.83·14-s − 15-s + 2.16·16-s − 6.88·17-s − 0.567·18-s + 4.58·19-s − 1.67·20-s − 3.22·21-s − 0.134·22-s + 1.13·23-s − 2.08·24-s + 25-s + 1.04·26-s − 27-s − 5.41·28-s + ⋯ |
| L(s) = 1 | − 0.401·2-s − 0.577·3-s − 0.838·4-s + 0.447·5-s + 0.231·6-s + 1.21·7-s + 0.738·8-s + 0.333·9-s − 0.179·10-s + 0.0716·11-s + 0.484·12-s − 0.508·13-s − 0.489·14-s − 0.258·15-s + 0.542·16-s − 1.67·17-s − 0.133·18-s + 1.05·19-s − 0.375·20-s − 0.704·21-s − 0.0287·22-s + 0.237·23-s − 0.426·24-s + 0.200·25-s + 0.204·26-s − 0.192·27-s − 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.253415160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253415160\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.567T + 2T^{2} \) |
| 7 | \( 1 - 3.22T + 7T^{2} \) |
| 11 | \( 1 - 0.237T + 11T^{2} \) |
| 13 | \( 1 + 1.83T + 13T^{2} \) |
| 17 | \( 1 + 6.88T + 17T^{2} \) |
| 19 | \( 1 - 4.58T + 19T^{2} \) |
| 23 | \( 1 - 1.13T + 23T^{2} \) |
| 29 | \( 1 - 9.72T + 29T^{2} \) |
| 31 | \( 1 + 8.43T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 0.213T + 43T^{2} \) |
| 47 | \( 1 + 3.76T + 47T^{2} \) |
| 53 | \( 1 + 0.170T + 53T^{2} \) |
| 59 | \( 1 - 3.17T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 0.857T + 67T^{2} \) |
| 71 | \( 1 - 5.94T + 71T^{2} \) |
| 73 | \( 1 + 7.28T + 73T^{2} \) |
| 79 | \( 1 + 6.65T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 8.45T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.161888221849992814655564152387, −7.43106003253575955256692461612, −6.77567274461024809160178823780, −5.78948093487968836873569785953, −5.05189665780104778546497883609, −4.68430126992194545802960756128, −3.93471348645717197308486979077, −2.55468947937274610416877519859, −1.60075834192750949594999066884, −0.68920322288190848344250073074,
0.68920322288190848344250073074, 1.60075834192750949594999066884, 2.55468947937274610416877519859, 3.93471348645717197308486979077, 4.68430126992194545802960756128, 5.05189665780104778546497883609, 5.78948093487968836873569785953, 6.77567274461024809160178823780, 7.43106003253575955256692461612, 8.161888221849992814655564152387
