| L(s) = 1 | − 2.40·2-s + 2.98·3-s + 3.77·4-s + 5-s − 7.17·6-s + 4.58·7-s − 4.25·8-s + 5.91·9-s − 2.40·10-s − 11-s + 11.2·12-s − 0.431·13-s − 11.0·14-s + 2.98·15-s + 2.68·16-s + 3.30·17-s − 14.2·18-s − 5.88·19-s + 3.77·20-s + 13.6·21-s + 2.40·22-s + 8.16·23-s − 12.7·24-s + 25-s + 1.03·26-s + 8.71·27-s + 17.3·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.72·3-s + 1.88·4-s + 0.447·5-s − 2.92·6-s + 1.73·7-s − 1.50·8-s + 1.97·9-s − 0.759·10-s − 0.301·11-s + 3.25·12-s − 0.119·13-s − 2.94·14-s + 0.771·15-s + 0.672·16-s + 0.800·17-s − 3.35·18-s − 1.35·19-s + 0.843·20-s + 2.98·21-s + 0.512·22-s + 1.70·23-s − 2.59·24-s + 0.200·25-s + 0.203·26-s + 1.67·27-s + 3.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.303854719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.303854719\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 2.40T + 2T^{2} \) |
| 3 | \( 1 - 2.98T + 3T^{2} \) |
| 7 | \( 1 - 4.58T + 7T^{2} \) |
| 13 | \( 1 + 0.431T + 13T^{2} \) |
| 17 | \( 1 - 3.30T + 17T^{2} \) |
| 19 | \( 1 + 5.88T + 19T^{2} \) |
| 23 | \( 1 - 8.16T + 23T^{2} \) |
| 29 | \( 1 + 7.11T + 29T^{2} \) |
| 31 | \( 1 + 7.86T + 31T^{2} \) |
| 37 | \( 1 + 1.45T + 37T^{2} \) |
| 41 | \( 1 - 2.34T + 41T^{2} \) |
| 43 | \( 1 - 6.71T + 43T^{2} \) |
| 47 | \( 1 + 2.26T + 47T^{2} \) |
| 53 | \( 1 - 1.59T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 1.99T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 79 | \( 1 + 0.232T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 - 6.70T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−8.588283240928028340053306060869, −7.82712514570734657921002680706, −7.57771031674034138330988390602, −6.85941274400956346584812171959, −5.52030277507329639103615815133, −4.57830225500539266723780862356, −3.49246201800915666448183270796, −2.32695115487943179575310025784, −1.97678388867564635942317905274, −1.12400583093353167831439265610,
1.12400583093353167831439265610, 1.97678388867564635942317905274, 2.32695115487943179575310025784, 3.49246201800915666448183270796, 4.57830225500539266723780862356, 5.52030277507329639103615815133, 6.85941274400956346584812171959, 7.57771031674034138330988390602, 7.82712514570734657921002680706, 8.588283240928028340053306060869
