| L(s) = 1 | − 1.76·2-s − 3-s + 1.12·4-s + 5-s + 1.76·6-s − 4.11·7-s + 1.55·8-s + 9-s − 1.76·10-s + 2.41·11-s − 1.12·12-s + 4.53·13-s + 7.26·14-s − 15-s − 4.98·16-s + 2.09·17-s − 1.76·18-s + 6.30·19-s + 1.12·20-s + 4.11·21-s − 4.25·22-s − 7.85·23-s − 1.55·24-s + 25-s − 8.01·26-s − 27-s − 4.61·28-s + ⋯ |
| L(s) = 1 | − 1.24·2-s − 0.577·3-s + 0.560·4-s + 0.447·5-s + 0.721·6-s − 1.55·7-s + 0.548·8-s + 0.333·9-s − 0.558·10-s + 0.726·11-s − 0.323·12-s + 1.25·13-s + 1.94·14-s − 0.258·15-s − 1.24·16-s + 0.507·17-s − 0.416·18-s + 1.44·19-s + 0.250·20-s + 0.897·21-s − 0.907·22-s − 1.63·23-s − 0.316·24-s + 0.200·25-s − 1.57·26-s − 0.192·27-s − 0.871·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8502260714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8502260714\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.76T + 2T^{2} \) |
| 7 | \( 1 + 4.11T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 - 2.09T + 17T^{2} \) |
| 19 | \( 1 - 6.30T + 19T^{2} \) |
| 23 | \( 1 + 7.85T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 - 6.13T + 31T^{2} \) |
| 37 | \( 1 - 2.57T + 37T^{2} \) |
| 41 | \( 1 - 8.16T + 41T^{2} \) |
| 43 | \( 1 + 2.19T + 43T^{2} \) |
| 47 | \( 1 - 7.63T + 47T^{2} \) |
| 53 | \( 1 - 4.49T + 53T^{2} \) |
| 59 | \( 1 + 8.39T + 59T^{2} \) |
| 61 | \( 1 - 5.83T + 61T^{2} \) |
| 67 | \( 1 - 3.90T + 67T^{2} \) |
| 71 | \( 1 + 8.06T + 71T^{2} \) |
| 73 | \( 1 - 3.87T + 73T^{2} \) |
| 79 | \( 1 - 3.65T + 79T^{2} \) |
| 83 | \( 1 + 8.55T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 4.27T + 97T^{2} \) |
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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
−8.150743831406040271023720080250, −7.45415392573016459636929209193, −6.58065096689058742373380505365, −6.21501023559384255648312566768, −5.55091591132244991391671595394, −4.32400484855186680377001422556, −3.62007879000595281007859397641, −2.61474236535023063894075129398, −1.30607399545700101795337138986, −0.70245887131594169362925635530,
0.70245887131594169362925635530, 1.30607399545700101795337138986, 2.61474236535023063894075129398, 3.62007879000595281007859397641, 4.32400484855186680377001422556, 5.55091591132244991391671595394, 6.21501023559384255648312566768, 6.58065096689058742373380505365, 7.45415392573016459636929209193, 8.150743831406040271023720080250
