| L(s) = 1 | − 2-s + 4-s − 5-s + 1.61·7-s − 8-s + 10-s + 5.85·11-s + 1.23·13-s − 1.61·14-s + 16-s + 5.61·17-s − 0.145·19-s − 20-s − 5.85·22-s + 4.47·23-s + 25-s − 1.23·26-s + 1.61·28-s + 5.70·29-s + 5.70·31-s − 32-s − 5.61·34-s − 1.61·35-s + 6.47·37-s + 0.145·38-s + 40-s + 5.38·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.611·7-s − 0.353·8-s + 0.316·10-s + 1.76·11-s + 0.342·13-s − 0.432·14-s + 0.250·16-s + 1.36·17-s − 0.0334·19-s − 0.223·20-s − 1.24·22-s + 0.932·23-s + 0.200·25-s − 0.242·26-s + 0.305·28-s + 1.05·29-s + 1.02·31-s − 0.176·32-s − 0.963·34-s − 0.273·35-s + 1.06·37-s + 0.0236·38-s + 0.158·40-s + 0.840·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.088398157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.088398157\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| good | 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 - 5.85T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 5.61T + 17T^{2} \) |
| 19 | \( 1 + 0.145T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 - 5.70T + 29T^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 - 5.38T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 1.85T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 5.70T + 59T^{2} \) |
| 61 | \( 1 + 5.32T + 61T^{2} \) |
| 67 | \( 1 + 3.61T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 1.52T + 73T^{2} \) |
| 79 | \( 1 + 2.09T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−7.895165723245932661729644664781, −7.30846091544533066465150730487, −6.48490188857405383225262400023, −6.03993929104132250422248720483, −4.96523645173634034338884798309, −4.24311476249203450135709271002, −3.46574351967327064259699876623, −2.64145528443223281672766344411, −1.30798552850389739123226365857, −0.982930974900222728984769009356,
0.982930974900222728984769009356, 1.30798552850389739123226365857, 2.64145528443223281672766344411, 3.46574351967327064259699876623, 4.24311476249203450135709271002, 4.96523645173634034338884798309, 6.03993929104132250422248720483, 6.48490188857405383225262400023, 7.30846091544533066465150730487, 7.895165723245932661729644664781
