| L(s) = 1 | − 2.69·2-s + 0.716·3-s + 5.28·4-s + 5-s − 1.93·6-s + 1.67·7-s − 8.85·8-s − 2.48·9-s − 2.69·10-s + 4.26·11-s + 3.78·12-s − 4.25·13-s − 4.51·14-s + 0.716·15-s + 13.3·16-s + 5.06·17-s + 6.71·18-s − 2.93·19-s + 5.28·20-s + 1.19·21-s − 11.5·22-s − 2.77·23-s − 6.34·24-s + 25-s + 11.4·26-s − 3.93·27-s + 8.83·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 0.413·3-s + 2.64·4-s + 0.447·5-s − 0.789·6-s + 0.632·7-s − 3.13·8-s − 0.828·9-s − 0.853·10-s + 1.28·11-s + 1.09·12-s − 1.18·13-s − 1.20·14-s + 0.184·15-s + 3.33·16-s + 1.22·17-s + 1.58·18-s − 0.672·19-s + 1.18·20-s + 0.261·21-s − 2.45·22-s − 0.579·23-s − 1.29·24-s + 0.200·25-s + 2.25·26-s − 0.756·27-s + 1.67·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 1201 | \( 1 - T \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 - 0.716T + 3T^{2} \) |
| 7 | \( 1 - 1.67T + 7T^{2} \) |
| 11 | \( 1 - 4.26T + 11T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 + 8.13T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 8.13T + 37T^{2} \) |
| 41 | \( 1 + 2.20T + 41T^{2} \) |
| 43 | \( 1 - 0.947T + 43T^{2} \) |
| 47 | \( 1 + 5.27T + 47T^{2} \) |
| 53 | \( 1 - 0.317T + 53T^{2} \) |
| 59 | \( 1 + 1.61T + 59T^{2} \) |
| 61 | \( 1 - 3.59T + 61T^{2} \) |
| 67 | \( 1 + 3.19T + 67T^{2} \) |
| 71 | \( 1 - 4.31T + 71T^{2} \) |
| 73 | \( 1 + 5.47T + 73T^{2} \) |
| 79 | \( 1 + 2.90T + 79T^{2} \) |
| 83 | \( 1 - 5.72T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.933285622380171235949838114844, −7.35058592696385986029779094374, −6.55273504716313620482003626101, −5.94597837882141267527787624763, −5.05261097210958683818061135927, −3.68719165972507015868364383326, −2.77201805258870070032912065754, −1.98659640558156588703874788242, −1.32070877085143862373882169685, 0,
1.32070877085143862373882169685, 1.98659640558156588703874788242, 2.77201805258870070032912065754, 3.68719165972507015868364383326, 5.05261097210958683818061135927, 5.94597837882141267527787624763, 6.55273504716313620482003626101, 7.35058592696385986029779094374, 7.933285622380171235949838114844
