| L(s) = 1 | + 2.48·2-s − 3-s + 4.15·4-s − 5-s − 2.48·6-s + 4.39·7-s + 5.33·8-s + 9-s − 2.48·10-s − 4.15·12-s + 2.13·13-s + 10.8·14-s + 15-s + 4.92·16-s − 3.96·17-s + 2.48·18-s + 2.15·19-s − 4.15·20-s − 4.39·21-s + 5.16·23-s − 5.33·24-s + 25-s + 5.30·26-s − 27-s + 18.2·28-s − 7.93·29-s + 2.48·30-s + ⋯ |
| L(s) = 1 | + 1.75·2-s − 0.577·3-s + 2.07·4-s − 0.447·5-s − 1.01·6-s + 1.66·7-s + 1.88·8-s + 0.333·9-s − 0.784·10-s − 1.19·12-s + 0.592·13-s + 2.91·14-s + 0.258·15-s + 1.23·16-s − 0.960·17-s + 0.584·18-s + 0.493·19-s − 0.928·20-s − 0.959·21-s + 1.07·23-s − 1.08·24-s + 0.200·25-s + 1.03·26-s − 0.192·27-s + 3.44·28-s − 1.47·29-s + 0.452·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.746905395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.746905395\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.48T + 2T^{2} \) |
| 7 | \( 1 - 4.39T + 7T^{2} \) |
| 13 | \( 1 - 2.13T + 13T^{2} \) |
| 17 | \( 1 + 3.96T + 17T^{2} \) |
| 19 | \( 1 - 2.15T + 19T^{2} \) |
| 23 | \( 1 - 5.16T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 + 8.57T + 37T^{2} \) |
| 41 | \( 1 - 7.63T + 41T^{2} \) |
| 43 | \( 1 - 2.22T + 43T^{2} \) |
| 47 | \( 1 - 6.88T + 47T^{2} \) |
| 53 | \( 1 + 5.69T + 53T^{2} \) |
| 59 | \( 1 - 3.78T + 59T^{2} \) |
| 61 | \( 1 + 4.83T + 61T^{2} \) |
| 67 | \( 1 - 9.57T + 67T^{2} \) |
| 71 | \( 1 + 2.11T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 3.26T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 7.64T + 89T^{2} \) |
| 97 | \( 1 + 5.67T + 97T^{2} \) |
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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
−9.185903782246689204547663847173, −8.212072931637460373192906693008, −7.36054014831588814383937273340, −6.70926194223653779077236606142, −5.67218186946419842940241942556, −5.11553052583674043942446245898, −4.42674382107166998849966934135, −3.76747804988242374666150396607, −2.49613713349009486147949459230, −1.37495595317750642924356249765,
1.37495595317750642924356249765, 2.49613713349009486147949459230, 3.76747804988242374666150396607, 4.42674382107166998849966934135, 5.11553052583674043942446245898, 5.67218186946419842940241942556, 6.70926194223653779077236606142, 7.36054014831588814383937273340, 8.212072931637460373192906693008, 9.185903782246689204547663847173
