| L(s) = 1 | − 0.806·3-s + 3.15·5-s − 0.675·7-s − 2.35·9-s − 2.51·11-s − 6.35·13-s − 2.54·15-s − 4.15·17-s + 0.418·19-s + 0.544·21-s − 3.48·23-s + 4.96·25-s + 4.31·27-s − 2.15·29-s − 6.73·31-s + 2.03·33-s − 2.13·35-s − 2.41·37-s + 5.11·39-s + 7.70·41-s + 8.15·43-s − 7.41·45-s + 11.6·47-s − 6.54·49-s + 3.35·51-s + 2.96·53-s − 7.95·55-s + ⋯ |
| L(s) = 1 | − 0.465·3-s + 1.41·5-s − 0.255·7-s − 0.783·9-s − 0.759·11-s − 1.76·13-s − 0.656·15-s − 1.00·17-s + 0.0959·19-s + 0.118·21-s − 0.725·23-s + 0.992·25-s + 0.829·27-s − 0.400·29-s − 1.20·31-s + 0.353·33-s − 0.360·35-s − 0.397·37-s + 0.819·39-s + 1.20·41-s + 1.24·43-s − 1.10·45-s + 1.70·47-s − 0.934·49-s + 0.469·51-s + 0.406·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 3 | \( 1 + 0.806T + 3T^{2} \) |
| 5 | \( 1 - 3.15T + 5T^{2} \) |
| 7 | \( 1 + 0.675T + 7T^{2} \) |
| 11 | \( 1 + 2.51T + 11T^{2} \) |
| 13 | \( 1 + 6.35T + 13T^{2} \) |
| 17 | \( 1 + 4.15T + 17T^{2} \) |
| 19 | \( 1 - 0.418T + 19T^{2} \) |
| 23 | \( 1 + 3.48T + 23T^{2} \) |
| 29 | \( 1 + 2.15T + 29T^{2} \) |
| 31 | \( 1 + 6.73T + 31T^{2} \) |
| 37 | \( 1 + 2.41T + 37T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 - 8.15T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 2.96T + 53T^{2} \) |
| 59 | \( 1 + 15.2T + 59T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 2.73T + 71T^{2} \) |
| 73 | \( 1 - 6.50T + 73T^{2} \) |
| 79 | \( 1 + 6.70T + 79T^{2} \) |
| 83 | \( 1 - 3.22T + 83T^{2} \) |
| 89 | \( 1 - 1.73T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−9.490946970128143621655762239848, −9.129853124780476410026747815070, −7.81538179746200291367222952722, −6.94283191202983046517948798218, −5.88328455629473274464052515867, −5.49452357487804324141162731456, −4.50435664086380839237012599150, −2.76240588619606825967672092657, −2.11220437557498441898351282824, 0,
2.11220437557498441898351282824, 2.76240588619606825967672092657, 4.50435664086380839237012599150, 5.49452357487804324141162731456, 5.88328455629473274464052515867, 6.94283191202983046517948798218, 7.81538179746200291367222952722, 9.129853124780476410026747815070, 9.490946970128143621655762239848
