| L(s) = 1 | + 0.619·3-s + 2.60·5-s − 7-s − 2.61·9-s + 4.07·11-s + 1.61·15-s − 7.64·17-s + 3.87·19-s − 0.619·21-s − 7.02·23-s + 1.76·25-s − 3.47·27-s − 4.09·29-s − 3.74·31-s + 2.52·33-s − 2.60·35-s + 8.83·37-s − 1.62·41-s + 3.25·43-s − 6.80·45-s + 8.01·47-s + 49-s − 4.73·51-s + 9.25·53-s + 10.6·55-s + 2.39·57-s − 6.25·59-s + ⋯ |
| L(s) = 1 | + 0.357·3-s + 1.16·5-s − 0.377·7-s − 0.872·9-s + 1.22·11-s + 0.415·15-s − 1.85·17-s + 0.887·19-s − 0.135·21-s − 1.46·23-s + 0.353·25-s − 0.669·27-s − 0.760·29-s − 0.672·31-s + 0.439·33-s − 0.439·35-s + 1.45·37-s − 0.254·41-s + 0.496·43-s − 1.01·45-s + 1.16·47-s + 0.142·49-s − 0.663·51-s + 1.27·53-s + 1.43·55-s + 0.317·57-s − 0.814·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.619T + 3T^{2} \) |
| 5 | \( 1 - 2.60T + 5T^{2} \) |
| 11 | \( 1 - 4.07T + 11T^{2} \) |
| 17 | \( 1 + 7.64T + 17T^{2} \) |
| 19 | \( 1 - 3.87T + 19T^{2} \) |
| 23 | \( 1 + 7.02T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 + 3.74T + 31T^{2} \) |
| 37 | \( 1 - 8.83T + 37T^{2} \) |
| 41 | \( 1 + 1.62T + 41T^{2} \) |
| 43 | \( 1 - 3.25T + 43T^{2} \) |
| 47 | \( 1 - 8.01T + 47T^{2} \) |
| 53 | \( 1 - 9.25T + 53T^{2} \) |
| 59 | \( 1 + 6.25T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 9.24T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 2.59T + 73T^{2} \) |
| 79 | \( 1 + 0.869T + 79T^{2} \) |
| 83 | \( 1 - 6.34T + 83T^{2} \) |
| 89 | \( 1 + 0.774T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.28734641994924355697749867413, −6.55671764276648236946910344157, −5.86879096831572917364268561772, −5.69011178481665836299745446613, −4.40383384450206349632729304785, −3.88583553860153755093866870554, −2.84720173542458002664471993369, −2.22324532974502126030361785447, −1.45825339042896232573382873108, 0,
1.45825339042896232573382873108, 2.22324532974502126030361785447, 2.84720173542458002664471993369, 3.88583553860153755093866870554, 4.40383384450206349632729304785, 5.69011178481665836299745446613, 5.86879096831572917364268561772, 6.55671764276648236946910344157, 7.28734641994924355697749867413
