| L(s) = 1 | − 0.912·3-s + 1.66·5-s − 2.16·9-s − 0.242·11-s + 5.76·13-s − 1.51·15-s − 5.76·17-s − 19-s + 6.61·23-s − 2.22·25-s + 4.71·27-s + 8.91·29-s + 5.93·31-s + 0.220·33-s − 9.80·37-s − 5.26·39-s + 2·41-s − 7.83·43-s − 3.60·45-s − 1.57·47-s + 5.26·51-s + 6.48·53-s − 0.403·55-s + 0.912·57-s − 1.07·59-s − 9.10·61-s + 9.60·65-s + ⋯ |
| L(s) = 1 | − 0.526·3-s + 0.744·5-s − 0.722·9-s − 0.0730·11-s + 1.60·13-s − 0.392·15-s − 1.39·17-s − 0.229·19-s + 1.37·23-s − 0.445·25-s + 0.907·27-s + 1.65·29-s + 1.06·31-s + 0.0384·33-s − 1.61·37-s − 0.842·39-s + 0.312·41-s − 1.19·43-s − 0.538·45-s − 0.230·47-s + 0.736·51-s + 0.890·53-s − 0.0543·55-s + 0.120·57-s − 0.139·59-s − 1.16·61-s + 1.19·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.799364055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.799364055\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.912T + 3T^{2} \) |
| 5 | \( 1 - 1.66T + 5T^{2} \) |
| 11 | \( 1 + 0.242T + 11T^{2} \) |
| 13 | \( 1 - 5.76T + 13T^{2} \) |
| 17 | \( 1 + 5.76T + 17T^{2} \) |
| 23 | \( 1 - 6.61T + 23T^{2} \) |
| 29 | \( 1 - 8.91T + 29T^{2} \) |
| 31 | \( 1 - 5.93T + 31T^{2} \) |
| 37 | \( 1 + 9.80T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 7.83T + 43T^{2} \) |
| 47 | \( 1 + 1.57T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 + 1.07T + 59T^{2} \) |
| 61 | \( 1 + 9.10T + 61T^{2} \) |
| 67 | \( 1 - 8.02T + 67T^{2} \) |
| 71 | \( 1 - 1.22T + 71T^{2} \) |
| 73 | \( 1 + 0.801T + 73T^{2} \) |
| 79 | \( 1 + 7.58T + 79T^{2} \) |
| 83 | \( 1 - 8.91T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 1.11T + 97T^{2} \) |
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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
−8.087409985320022838976574715473, −6.79122382058790872527964133478, −6.50772120930892124391520114098, −5.90644944119397289311334051955, −5.13053673650357992531298359292, −4.52693380132124057693834249887, −3.45605760346643749796850925361, −2.69888899248272404534214859181, −1.72671915796987837296735388566, −0.70334340376276612674475086896,
0.70334340376276612674475086896, 1.72671915796987837296735388566, 2.69888899248272404534214859181, 3.45605760346643749796850925361, 4.52693380132124057693834249887, 5.13053673650357992531298359292, 5.90644944119397289311334051955, 6.50772120930892124391520114098, 6.79122382058790872527964133478, 8.087409985320022838976574715473
