| L(s) = 1 | + 0.868·3-s + 0.636·5-s − 2.24·9-s − 2.24·11-s + 1.27·13-s + 0.553·15-s + 7.52·17-s + 5.80·19-s − 8.28·23-s − 4.59·25-s − 4.55·27-s − 9.88·29-s − 3.59·31-s − 1.94·33-s + 0.798·37-s + 1.10·39-s − 10.5·41-s + 5.28·43-s − 1.42·45-s − 9.65·47-s + 6.53·51-s + 6.48·53-s − 1.42·55-s + 5.04·57-s − 12.8·59-s + 6.95·61-s + 0.812·65-s + ⋯ |
| L(s) = 1 | + 0.501·3-s + 0.284·5-s − 0.748·9-s − 0.675·11-s + 0.353·13-s + 0.142·15-s + 1.82·17-s + 1.33·19-s − 1.72·23-s − 0.918·25-s − 0.877·27-s − 1.83·29-s − 0.646·31-s − 0.339·33-s + 0.131·37-s + 0.177·39-s − 1.64·41-s + 0.806·43-s − 0.213·45-s − 1.40·47-s + 0.915·51-s + 0.890·53-s − 0.192·55-s + 0.668·57-s − 1.67·59-s + 0.890·61-s + 0.100·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.868T + 3T^{2} \) |
| 5 | \( 1 - 0.636T + 5T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 - 1.27T + 13T^{2} \) |
| 17 | \( 1 - 7.52T + 17T^{2} \) |
| 19 | \( 1 - 5.80T + 19T^{2} \) |
| 23 | \( 1 + 8.28T + 23T^{2} \) |
| 29 | \( 1 + 9.88T + 29T^{2} \) |
| 31 | \( 1 + 3.59T + 31T^{2} \) |
| 37 | \( 1 - 0.798T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 5.28T + 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 6.95T + 61T^{2} \) |
| 67 | \( 1 - 2.48T + 67T^{2} \) |
| 71 | \( 1 + 4.09T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 5.40T + 79T^{2} \) |
| 83 | \( 1 + 6.73T + 83T^{2} \) |
| 89 | \( 1 + 5.01T + 89T^{2} \) |
| 97 | \( 1 - 8.96T + 97T^{2} \) |
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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
−7.77494131128282821876173303038, −7.46144452717487259939929429219, −6.14305336109965852308185585332, −5.60948849511356860973388163450, −5.17050202825289725247651225822, −3.65755331159078467861508116108, −3.46239664505678206841840157386, −2.35942889580493163324884040746, −1.51960091924318347505082813339, 0,
1.51960091924318347505082813339, 2.35942889580493163324884040746, 3.46239664505678206841840157386, 3.65755331159078467861508116108, 5.17050202825289725247651225822, 5.60948849511356860973388163450, 6.14305336109965852308185585332, 7.46144452717487259939929429219, 7.77494131128282821876173303038
