| L(s) = 1 | − 3-s + 4.27·5-s − 4·7-s + 9-s − 2.27·11-s + 4.27·13-s − 4.27·15-s + 2.27·19-s + 4·21-s − 2.27·23-s + 13.2·25-s − 27-s − 2·29-s + 8.54·31-s + 2.27·33-s − 17.0·35-s + 10.5·37-s − 4.27·39-s − 8.27·41-s − 10.2·43-s + 4.27·45-s + 4.54·47-s + 9·49-s − 10·53-s − 9.72·55-s − 2.27·57-s + 8.54·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.91·5-s − 1.51·7-s + 0.333·9-s − 0.685·11-s + 1.18·13-s − 1.10·15-s + 0.521·19-s + 0.872·21-s − 0.474·23-s + 2.65·25-s − 0.192·27-s − 0.371·29-s + 1.53·31-s + 0.396·33-s − 2.89·35-s + 1.73·37-s − 0.684·39-s − 1.29·41-s − 1.56·43-s + 0.637·45-s + 0.663·47-s + 1.28·49-s − 1.37·53-s − 1.31·55-s − 0.301·57-s + 1.11·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.049256679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.049256679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 4.27T + 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 13 | \( 1 - 4.27T + 13T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 + 2.27T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8.54T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 8.27T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 8.54T + 59T^{2} \) |
| 61 | \( 1 - 2.54T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 8.54T + 79T^{2} \) |
| 83 | \( 1 + 8.54T + 83T^{2} \) |
| 89 | \( 1 + 2.54T + 89T^{2} \) |
| 97 | \( 1 + 6.54T + 97T^{2} \) |
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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
−8.006494838061998329861854439869, −6.73087900998175329256423488185, −6.54776192595779720905754720419, −5.83912702137649442277808378735, −5.48212771199157144034155383156, −4.53354341962801026665918937529, −3.35966332081339539431449231984, −2.74359091075664300447444585216, −1.77247567342309774084514430307, −0.76115311641443121644798091966,
0.76115311641443121644798091966, 1.77247567342309774084514430307, 2.74359091075664300447444585216, 3.35966332081339539431449231984, 4.53354341962801026665918937529, 5.48212771199157144034155383156, 5.83912702137649442277808378735, 6.54776192595779720905754720419, 6.73087900998175329256423488185, 8.006494838061998329861854439869
