| L(s) = 1 | + 2.77·2-s + 3-s + 5.68·4-s − 5-s + 2.77·6-s − 2.27·7-s + 10.2·8-s + 9-s − 2.77·10-s + 5.68·12-s − 0.435·13-s − 6.31·14-s − 15-s + 16.9·16-s + 5·17-s + 2.77·18-s + 4.69·19-s − 5.68·20-s − 2.27·21-s − 0.845·23-s + 10.2·24-s + 25-s − 1.20·26-s + 27-s − 12.9·28-s − 2.65·29-s − 2.77·30-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 0.577·3-s + 2.84·4-s − 0.447·5-s + 1.13·6-s − 0.860·7-s + 3.61·8-s + 0.333·9-s − 0.876·10-s + 1.64·12-s − 0.120·13-s − 1.68·14-s − 0.258·15-s + 4.23·16-s + 1.21·17-s + 0.653·18-s + 1.07·19-s − 1.27·20-s − 0.497·21-s − 0.176·23-s + 2.08·24-s + 0.200·25-s − 0.236·26-s + 0.192·27-s − 2.44·28-s − 0.493·29-s − 0.506·30-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
| \(L(1)\) |
\(\approx\) |
\(6.774097596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.774097596\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.77T + 2T^{2} \) |
| 7 | \( 1 + 2.27T + 7T^{2} \) |
| 13 | \( 1 + 0.435T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - 4.69T + 19T^{2} \) |
| 23 | \( 1 + 0.845T + 23T^{2} \) |
| 29 | \( 1 + 2.65T + 29T^{2} \) |
| 31 | \( 1 + 4.66T + 31T^{2} \) |
| 37 | \( 1 + 8.86T + 37T^{2} \) |
| 41 | \( 1 + 4.29T + 41T^{2} \) |
| 43 | \( 1 - 7.00T + 43T^{2} \) |
| 47 | \( 1 - 0.468T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 3.93T + 59T^{2} \) |
| 61 | \( 1 - 2.96T + 61T^{2} \) |
| 67 | \( 1 + 2.47T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 8.42T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 5.92T + 83T^{2} \) |
| 89 | \( 1 - 5.89T + 89T^{2} \) |
| 97 | \( 1 + 8.64T + 97T^{2} \) |
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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
−9.436190879491210230120208214116, −8.085221858949638065619913187768, −7.35480781473594308143738570868, −6.82259105217181495350336224991, −5.78118167226277986645457025147, −5.19250724121233944564822032952, −4.10254823231023181156638006854, −3.38421111217664663995227810502, −2.93780336207530987485165782559, −1.61000857779018475543684992671,
1.61000857779018475543684992671, 2.93780336207530987485165782559, 3.38421111217664663995227810502, 4.10254823231023181156638006854, 5.19250724121233944564822032952, 5.78118167226277986645457025147, 6.82259105217181495350336224991, 7.35480781473594308143738570868, 8.085221858949638065619913187768, 9.436190879491210230120208214116
