| L(s) = 1 | + 2-s + 4-s − 2.63·5-s − 2.24·7-s + 8-s − 2.63·10-s − 4.09·11-s + 0.00776·13-s − 2.24·14-s + 16-s + 0.673·17-s − 3.57·19-s − 2.63·20-s − 4.09·22-s + 3.39·23-s + 1.93·25-s + 0.00776·26-s − 2.24·28-s + 8.00·29-s − 6.93·31-s + 32-s + 0.673·34-s + 5.91·35-s + 0.651·37-s − 3.57·38-s − 2.63·40-s + 4.43·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.17·5-s − 0.849·7-s + 0.353·8-s − 0.832·10-s − 1.23·11-s + 0.00215·13-s − 0.600·14-s + 0.250·16-s + 0.163·17-s − 0.819·19-s − 0.588·20-s − 0.872·22-s + 0.707·23-s + 0.386·25-s + 0.00152·26-s − 0.424·28-s + 1.48·29-s − 1.24·31-s + 0.176·32-s + 0.115·34-s + 0.999·35-s + 0.107·37-s − 0.579·38-s − 0.416·40-s + 0.692·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.567245560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.567245560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 2.63T + 5T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 + 4.09T + 11T^{2} \) |
| 13 | \( 1 - 0.00776T + 13T^{2} \) |
| 17 | \( 1 - 0.673T + 17T^{2} \) |
| 19 | \( 1 + 3.57T + 19T^{2} \) |
| 23 | \( 1 - 3.39T + 23T^{2} \) |
| 29 | \( 1 - 8.00T + 29T^{2} \) |
| 31 | \( 1 + 6.93T + 31T^{2} \) |
| 37 | \( 1 - 0.651T + 37T^{2} \) |
| 41 | \( 1 - 4.43T + 41T^{2} \) |
| 43 | \( 1 - 9.59T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 4.64T + 53T^{2} \) |
| 59 | \( 1 - 7.95T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 9.64T + 71T^{2} \) |
| 73 | \( 1 - 0.995T + 73T^{2} \) |
| 79 | \( 1 + 1.27T + 79T^{2} \) |
| 83 | \( 1 - 5.75T + 83T^{2} \) |
| 89 | \( 1 + 2.07T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−8.298365979222595031565770574730, −7.54065735185732630306952775286, −7.07577805338556147922076568622, −6.15998193606196826167066382990, −5.43413417563196035473790593079, −4.53429804550446122013146222814, −3.89532713294985366444561731618, −3.06435746579956241795477528101, −2.38050285089266399674421993188, −0.61351468390173504201455675631,
0.61351468390173504201455675631, 2.38050285089266399674421993188, 3.06435746579956241795477528101, 3.89532713294985366444561731618, 4.53429804550446122013146222814, 5.43413417563196035473790593079, 6.15998193606196826167066382990, 7.07577805338556147922076568622, 7.54065735185732630306952775286, 8.298365979222595031565770574730
