| L(s) = 1 | − 2.71·2-s − 1.56·3-s + 5.35·4-s + 1.41·5-s + 4.23·6-s − 1.24·7-s − 9.11·8-s − 0.564·9-s − 3.84·10-s − 1.09·11-s − 8.36·12-s − 1.70·13-s + 3.36·14-s − 2.21·15-s + 14.0·16-s + 0.0690·17-s + 1.53·18-s − 5.12·19-s + 7.60·20-s + 1.93·21-s + 2.95·22-s + 5.50·23-s + 14.2·24-s − 2.98·25-s + 4.62·26-s + 5.56·27-s − 6.64·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 0.901·3-s + 2.67·4-s + 0.634·5-s + 1.72·6-s − 0.468·7-s − 3.22·8-s − 0.188·9-s − 1.21·10-s − 0.328·11-s − 2.41·12-s − 0.472·13-s + 0.899·14-s − 0.571·15-s + 3.50·16-s + 0.0167·17-s + 0.360·18-s − 1.17·19-s + 1.69·20-s + 0.422·21-s + 0.630·22-s + 1.14·23-s + 2.90·24-s − 0.597·25-s + 0.906·26-s + 1.07·27-s − 1.25·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2346231230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2346231230\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.71T + 2T^{2} \) |
| 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 - 0.0690T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 - 5.50T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 + 1.98T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 2.16T + 41T^{2} \) |
| 43 | \( 1 + 7.04T + 43T^{2} \) |
| 47 | \( 1 + 6.34T + 47T^{2} \) |
| 53 | \( 1 + 9.99T + 53T^{2} \) |
| 59 | \( 1 + 7.58T + 59T^{2} \) |
| 61 | \( 1 + 7.37T + 61T^{2} \) |
| 67 | \( 1 - 7.07T + 67T^{2} \) |
| 71 | \( 1 - 5.26T + 71T^{2} \) |
| 73 | \( 1 - 8.99T + 73T^{2} \) |
| 79 | \( 1 - 4.05T + 79T^{2} \) |
| 83 | \( 1 - 1.64T + 83T^{2} \) |
| 89 | \( 1 - 8.31T + 89T^{2} \) |
| 97 | \( 1 + 7.87T + 97T^{2} \) |
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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
−8.183759360186699542350233229144, −7.76182323374325760290745347784, −6.61344933795254148249358059353, −6.49869475369615393875965083595, −5.70128276581111558025982709549, −4.86755373676768049140597690312, −3.26551089087587380502126839833, −2.43556731381829002946412980925, −1.58523169219992779609272723961, −0.36259107103742167078030915259,
0.36259107103742167078030915259, 1.58523169219992779609272723961, 2.43556731381829002946412980925, 3.26551089087587380502126839833, 4.86755373676768049140597690312, 5.70128276581111558025982709549, 6.49869475369615393875965083595, 6.61344933795254148249358059353, 7.76182323374325760290745347784, 8.183759360186699542350233229144
