| L(s) = 1 | + 3.18·3-s − 0.628·5-s − 2.91·7-s + 7.11·9-s + 1.79·11-s − 0.925·13-s − 1.99·15-s − 2.99·17-s + 2.66·19-s − 9.28·21-s − 5.21·23-s − 4.60·25-s + 13.0·27-s − 6.23·29-s − 8.37·31-s + 5.71·33-s + 1.83·35-s − 10.3·37-s − 2.94·39-s + 7.74·41-s − 12.2·43-s − 4.47·45-s + 2.78·47-s + 1.51·49-s − 9.51·51-s + 8.90·53-s − 1.12·55-s + ⋯ |
| L(s) = 1 | + 1.83·3-s − 0.281·5-s − 1.10·7-s + 2.37·9-s + 0.541·11-s − 0.256·13-s − 0.516·15-s − 0.725·17-s + 0.610·19-s − 2.02·21-s − 1.08·23-s − 0.920·25-s + 2.52·27-s − 1.15·29-s − 1.50·31-s + 0.995·33-s + 0.309·35-s − 1.70·37-s − 0.471·39-s + 1.20·41-s − 1.86·43-s − 0.666·45-s + 0.406·47-s + 0.215·49-s − 1.33·51-s + 1.22·53-s − 0.152·55-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 | \( 1 - 3.18T + 3T^{2} \) |
| 5 | \( 1 + 0.628T + 5T^{2} \) |
| 7 | \( 1 + 2.91T + 7T^{2} \) |
| 11 | \( 1 - 1.79T + 11T^{2} \) |
| 13 | \( 1 + 0.925T + 13T^{2} \) |
| 17 | \( 1 + 2.99T + 17T^{2} \) |
| 19 | \( 1 - 2.66T + 19T^{2} \) |
| 23 | \( 1 + 5.21T + 23T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 + 8.37T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 7.74T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 2.78T + 47T^{2} \) |
| 53 | \( 1 - 8.90T + 53T^{2} \) |
| 59 | \( 1 - 1.47T + 59T^{2} \) |
| 61 | \( 1 + 0.606T + 61T^{2} \) |
| 67 | \( 1 + 1.11T + 67T^{2} \) |
| 71 | \( 1 - 4.19T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 5.84T + 79T^{2} \) |
| 83 | \( 1 + 9.83T + 83T^{2} \) |
| 89 | \( 1 - 1.16T + 89T^{2} \) |
| 97 | \( 1 - 1.30T + 97T^{2} \) |
| 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
−7.41728491091598110837267063003, −7.15536700012976614382250255988, −6.30783170170026375362117457504, −5.37620175363322975314169464989, −4.19161222213774292695624013021, −3.70221064345992015576021909554, −3.26867749190933164491877525287, −2.26912993691449539454549453872, −1.68373105930503822335073725683, 0,
1.68373105930503822335073725683, 2.26912993691449539454549453872, 3.26867749190933164491877525287, 3.70221064345992015576021909554, 4.19161222213774292695624013021, 5.37620175363322975314169464989, 6.30783170170026375362117457504, 7.15536700012976614382250255988, 7.41728491091598110837267063003
