| L(s) = 1 | − 3-s − 0.850·5-s + 1.73·7-s + 9-s + 0.361·11-s + 4.60·13-s + 0.850·15-s − 1.44·19-s − 1.73·21-s + 1.05·23-s − 4.27·25-s − 27-s − 2.88·29-s − 4.17·31-s − 0.361·33-s − 1.47·35-s − 6.58·37-s − 4.60·39-s − 4.23·41-s − 5.22·43-s − 0.850·45-s − 2.60·47-s − 3.99·49-s + 1.61·53-s − 0.307·55-s + 1.44·57-s + 10.9·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.380·5-s + 0.655·7-s + 0.333·9-s + 0.109·11-s + 1.27·13-s + 0.219·15-s − 0.331·19-s − 0.378·21-s + 0.219·23-s − 0.855·25-s − 0.192·27-s − 0.534·29-s − 0.749·31-s − 0.0629·33-s − 0.249·35-s − 1.08·37-s − 0.737·39-s − 0.661·41-s − 0.796·43-s − 0.126·45-s − 0.379·47-s − 0.570·49-s + 0.221·53-s − 0.0415·55-s + 0.191·57-s + 1.42·59-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 0.850T + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 - 0.361T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 19 | \( 1 + 1.44T + 19T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 + 2.88T + 29T^{2} \) |
| 31 | \( 1 + 4.17T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 + 5.22T + 43T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 - 1.61T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 9.62T + 61T^{2} \) |
| 67 | \( 1 - 8.12T + 67T^{2} \) |
| 71 | \( 1 - 1.13T + 71T^{2} \) |
| 73 | \( 1 - 3.95T + 73T^{2} \) |
| 79 | \( 1 - 1.86T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.60888042278600510277150025695, −6.85936792172869664106517934840, −6.20478086952835101001241386935, −5.45156665354287318402377777428, −4.84689652015228420574588887643, −3.90498262246230633721141415271, −3.45631622667735741308635650223, −2.03483972912618317564268105069, −1.29686502255034160320028069763, 0,
1.29686502255034160320028069763, 2.03483972912618317564268105069, 3.45631622667735741308635650223, 3.90498262246230633721141415271, 4.84689652015228420574588887643, 5.45156665354287318402377777428, 6.20478086952835101001241386935, 6.85936792172869664106517934840, 7.60888042278600510277150025695
