| L(s) = 1 | − 2.27·3-s + 1.62·5-s − 3.69·7-s + 2.19·9-s − 5.39·11-s − 0.869·13-s − 3.69·15-s − 3.44·17-s + 5.99·19-s + 8.43·21-s − 2.36·25-s + 1.82·27-s + 6.59·29-s − 6.66·31-s + 12.2·33-s − 6.00·35-s + 1.89·37-s + 1.98·39-s + 8.09·41-s + 12.3·43-s + 3.56·45-s + 3.55·47-s + 6.68·49-s + 7.85·51-s + 8.82·53-s − 8.75·55-s − 13.6·57-s + ⋯ |
| L(s) = 1 | − 1.31·3-s + 0.725·5-s − 1.39·7-s + 0.732·9-s − 1.62·11-s − 0.241·13-s − 0.955·15-s − 0.835·17-s + 1.37·19-s + 1.84·21-s − 0.473·25-s + 0.351·27-s + 1.22·29-s − 1.19·31-s + 2.14·33-s − 1.01·35-s + 0.311·37-s + 0.317·39-s + 1.26·41-s + 1.88·43-s + 0.531·45-s + 0.518·47-s + 0.955·49-s + 1.09·51-s + 1.21·53-s − 1.17·55-s − 1.80·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.27T + 3T^{2} \) |
| 5 | \( 1 - 1.62T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 + 5.39T + 11T^{2} \) |
| 13 | \( 1 + 0.869T + 13T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 19 | \( 1 - 5.99T + 19T^{2} \) |
| 29 | \( 1 - 6.59T + 29T^{2} \) |
| 31 | \( 1 + 6.66T + 31T^{2} \) |
| 37 | \( 1 - 1.89T + 37T^{2} \) |
| 41 | \( 1 - 8.09T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 - 3.55T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 + 7.08T + 67T^{2} \) |
| 71 | \( 1 + 0.0791T + 71T^{2} \) |
| 73 | \( 1 - 3.26T + 73T^{2} \) |
| 79 | \( 1 - 9.31T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 2.62T + 89T^{2} \) |
| 97 | \( 1 + 3.58T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.29991498513920080934608635874, −6.55801409234022879651890440384, −5.94727759155831284313814828859, −5.53047634538012507363960697114, −4.95533673764185273598693567460, −3.98725327773989230937879933766, −2.85920937447918190692691819739, −2.39820222175631320187018973856, −0.878988879871907832555873084771, 0,
0.878988879871907832555873084771, 2.39820222175631320187018973856, 2.85920937447918190692691819739, 3.98725327773989230937879933766, 4.95533673764185273598693567460, 5.53047634538012507363960697114, 5.94727759155831284313814828859, 6.55801409234022879651890440384, 7.29991498513920080934608635874
