| L(s) = 1 | − 1.46·2-s + 3-s + 0.157·4-s − 5-s − 1.46·6-s − 1.36·7-s + 2.70·8-s + 9-s + 1.46·10-s + 0.919·11-s + 0.157·12-s − 2.49·13-s + 1.99·14-s − 15-s − 4.28·16-s − 0.335·17-s − 1.46·18-s − 2.84·19-s − 0.157·20-s − 1.36·21-s − 1.35·22-s + 6.04·23-s + 2.70·24-s + 25-s + 3.66·26-s + 27-s − 0.213·28-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 0.577·3-s + 0.0785·4-s − 0.447·5-s − 0.599·6-s − 0.514·7-s + 0.956·8-s + 0.333·9-s + 0.464·10-s + 0.277·11-s + 0.0453·12-s − 0.692·13-s + 0.534·14-s − 0.258·15-s − 1.07·16-s − 0.0812·17-s − 0.346·18-s − 0.652·19-s − 0.0351·20-s − 0.297·21-s − 0.288·22-s + 1.26·23-s + 0.552·24-s + 0.200·25-s + 0.719·26-s + 0.192·27-s − 0.0403·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8688442833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8688442833\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.46T + 2T^{2} \) |
| 7 | \( 1 + 1.36T + 7T^{2} \) |
| 11 | \( 1 - 0.919T + 11T^{2} \) |
| 13 | \( 1 + 2.49T + 13T^{2} \) |
| 17 | \( 1 + 0.335T + 17T^{2} \) |
| 19 | \( 1 + 2.84T + 19T^{2} \) |
| 23 | \( 1 - 6.04T + 23T^{2} \) |
| 29 | \( 1 - 0.957T + 29T^{2} \) |
| 31 | \( 1 - 9.05T + 31T^{2} \) |
| 37 | \( 1 + 5.39T + 37T^{2} \) |
| 41 | \( 1 + 3.36T + 41T^{2} \) |
| 43 | \( 1 + 2.20T + 43T^{2} \) |
| 47 | \( 1 - 6.54T + 47T^{2} \) |
| 53 | \( 1 + 2.32T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 2.16T + 61T^{2} \) |
| 67 | \( 1 + 7.99T + 67T^{2} \) |
| 71 | \( 1 - 2.00T + 71T^{2} \) |
| 73 | \( 1 - 7.50T + 73T^{2} \) |
| 79 | \( 1 - 8.79T + 79T^{2} \) |
| 83 | \( 1 + 4.52T + 83T^{2} \) |
| 89 | \( 1 + 5.84T + 89T^{2} \) |
| 97 | \( 1 - 3.98T + 97T^{2} \) |
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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
−8.264356474258972139429196843909, −7.51534004870273272742026947406, −6.95974899024381331832203142441, −6.28194077972764894344432399845, −4.92998154370969045140261448899, −4.50065447585759001989307315606, −3.50158504203061697125105012978, −2.71597608903724751679662246456, −1.65874825964345858614853476221, −0.56867851531596136533352462962,
0.56867851531596136533352462962, 1.65874825964345858614853476221, 2.71597608903724751679662246456, 3.50158504203061697125105012978, 4.50065447585759001989307315606, 4.92998154370969045140261448899, 6.28194077972764894344432399845, 6.95974899024381331832203142441, 7.51534004870273272742026947406, 8.264356474258972139429196843909
