| L(s) = 1 | + 1.19·3-s − 0.449·5-s + 2.62·7-s − 1.57·9-s + 0.634·11-s + 0.888·13-s − 0.536·15-s − 4.11·17-s + 7.30·19-s + 3.12·21-s + 3.95·23-s − 4.79·25-s − 5.45·27-s − 9.40·29-s − 10.3·31-s + 0.756·33-s − 1.18·35-s − 3.75·37-s + 1.05·39-s − 10.1·41-s + 6.79·43-s + 0.710·45-s − 4.84·47-s − 0.110·49-s − 4.89·51-s − 7.23·53-s − 0.285·55-s + ⋯ |
| L(s) = 1 | + 0.688·3-s − 0.201·5-s + 0.992·7-s − 0.526·9-s + 0.191·11-s + 0.246·13-s − 0.138·15-s − 0.996·17-s + 1.67·19-s + 0.682·21-s + 0.824·23-s − 0.959·25-s − 1.05·27-s − 1.74·29-s − 1.85·31-s + 0.131·33-s − 0.199·35-s − 0.617·37-s + 0.169·39-s − 1.59·41-s + 1.03·43-s + 0.105·45-s − 0.706·47-s − 0.0157·49-s − 0.686·51-s − 0.993·53-s − 0.0385·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 + 0.449T + 5T^{2} \) |
| 7 | \( 1 - 2.62T + 7T^{2} \) |
| 11 | \( 1 - 0.634T + 11T^{2} \) |
| 13 | \( 1 - 0.888T + 13T^{2} \) |
| 17 | \( 1 + 4.11T + 17T^{2} \) |
| 19 | \( 1 - 7.30T + 19T^{2} \) |
| 23 | \( 1 - 3.95T + 23T^{2} \) |
| 29 | \( 1 + 9.40T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 3.75T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 6.79T + 43T^{2} \) |
| 47 | \( 1 + 4.84T + 47T^{2} \) |
| 53 | \( 1 + 7.23T + 53T^{2} \) |
| 59 | \( 1 + 8.29T + 59T^{2} \) |
| 61 | \( 1 - 5.40T + 61T^{2} \) |
| 67 | \( 1 - 5.99T + 67T^{2} \) |
| 71 | \( 1 + 3.95T + 71T^{2} \) |
| 73 | \( 1 + 8.81T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 8.50T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.51008044559903965843892030152, −7.13214122075828265852355384349, −5.96017576061278636188015904630, −5.36338713965385551295740434764, −4.70719843423986580158542186850, −3.64316970866596440774529696886, −3.29722740397989032292796310972, −2.10273489538232935328213899856, −1.54489796228155370780668931301, 0,
1.54489796228155370780668931301, 2.10273489538232935328213899856, 3.29722740397989032292796310972, 3.64316970866596440774529696886, 4.70719843423986580158542186850, 5.36338713965385551295740434764, 5.96017576061278636188015904630, 7.13214122075828265852355384349, 7.51008044559903965843892030152
