L(s) = 1 | − 2-s + 3-s + 4-s − 1.98·5-s − 6-s − 2.53·7-s − 8-s + 9-s + 1.98·10-s + 6.11·11-s + 12-s − 1.71·13-s + 2.53·14-s − 1.98·15-s + 16-s + 17-s − 18-s + 4.16·19-s − 1.98·20-s − 2.53·21-s − 6.11·22-s − 8.88·23-s − 24-s − 1.07·25-s + 1.71·26-s + 27-s − 2.53·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.886·5-s − 0.408·6-s − 0.958·7-s − 0.353·8-s + 0.333·9-s + 0.626·10-s + 1.84·11-s + 0.288·12-s − 0.475·13-s + 0.677·14-s − 0.511·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.956·19-s − 0.443·20-s − 0.553·21-s − 1.30·22-s − 1.85·23-s − 0.204·24-s − 0.214·25-s + 0.336·26-s + 0.192·27-s − 0.479·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 + 1.98T + 5T^{2} \) |
| 7 | \( 1 + 2.53T + 7T^{2} \) |
| 11 | \( 1 - 6.11T + 11T^{2} \) |
| 13 | \( 1 + 1.71T + 13T^{2} \) |
| 19 | \( 1 - 4.16T + 19T^{2} \) |
| 23 | \( 1 + 8.88T + 23T^{2} \) |
| 29 | \( 1 - 2.92T + 29T^{2} \) |
| 31 | \( 1 + 6.42T + 31T^{2} \) |
| 37 | \( 1 - 0.684T + 37T^{2} \) |
| 41 | \( 1 - 6.28T + 41T^{2} \) |
| 43 | \( 1 - 0.490T + 43T^{2} \) |
| 47 | \( 1 + 2.59T + 47T^{2} \) |
| 53 | \( 1 + 8.69T + 53T^{2} \) |
| 61 | \( 1 - 2.45T + 61T^{2} \) |
| 67 | \( 1 + 2.32T + 67T^{2} \) |
| 71 | \( 1 + 6.70T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 3.42T + 79T^{2} \) |
| 83 | \( 1 + 0.548T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 4.28T + 97T^{2} \) |
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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
−7.67011260166335765053084666141, −7.29603573546476119904430439413, −6.43682138768638689445677163660, −5.93539043906562017959425969685, −4.57808301919095337019137604943, −3.67699880365849438175712318401, −3.44309902658862931492857642597, −2.24844116542332086765954966305, −1.22520379055880673771344961768, 0,
1.22520379055880673771344961768, 2.24844116542332086765954966305, 3.44309902658862931492857642597, 3.67699880365849438175712318401, 4.57808301919095337019137604943, 5.93539043906562017959425969685, 6.43682138768638689445677163660, 7.29603573546476119904430439413, 7.67011260166335765053084666141