| L(s) = 1 | − 4·5-s − 2·11-s − 13-s − 2·17-s − 8·19-s + 4·23-s + 11·25-s + 6·29-s + 4·31-s + 6·37-s + 12·41-s − 4·43-s − 6·47-s − 7·49-s + 2·53-s + 8·55-s − 14·59-s + 10·61-s + 4·65-s + 4·67-s + 2·71-s − 2·73-s + 8·79-s + 14·83-s + 8·85-s + 32·95-s − 10·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.603·11-s − 0.277·13-s − 0.485·17-s − 1.83·19-s + 0.834·23-s + 11/5·25-s + 1.11·29-s + 0.718·31-s + 0.986·37-s + 1.87·41-s − 0.609·43-s − 0.875·47-s − 49-s + 0.274·53-s + 1.07·55-s − 1.82·59-s + 1.28·61-s + 0.496·65-s + 0.488·67-s + 0.237·71-s − 0.234·73-s + 0.900·79-s + 1.53·83-s + 0.867·85-s + 3.28·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8323398102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8323398102\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.006771409809014628468863978608, −8.251129103882234148491377458104, −7.86054071930096752093791369873, −6.93210531622838702609773257966, −6.24555617445133637550942476068, −4.72960594562341554496410559219, −4.46843413129605389307691420120, −3.37653352192825903166986925682, −2.44575355061988739951957175223, −0.59372633464493589187309274913,
0.59372633464493589187309274913, 2.44575355061988739951957175223, 3.37653352192825903166986925682, 4.46843413129605389307691420120, 4.72960594562341554496410559219, 6.24555617445133637550942476068, 6.93210531622838702609773257966, 7.86054071930096752093791369873, 8.251129103882234148491377458104, 9.006771409809014628468863978608
