L(s) = 1 | − 2.73·3-s − 5-s − 0.736·7-s + 4.45·9-s + 2.91·11-s + 1.97·13-s + 2.73·15-s + 4.98·17-s + 6.47·19-s + 2.01·21-s + 6.25·23-s + 25-s − 3.98·27-s − 4.32·29-s + 2.36·31-s − 7.95·33-s + 0.736·35-s + 4.11·37-s − 5.39·39-s − 0.914·41-s + 11.2·43-s − 4.45·45-s + 2.35·47-s − 6.45·49-s − 13.6·51-s + 6.79·53-s − 2.91·55-s + ⋯ |
L(s) = 1 | − 1.57·3-s − 0.447·5-s − 0.278·7-s + 1.48·9-s + 0.878·11-s + 0.547·13-s + 0.705·15-s + 1.20·17-s + 1.48·19-s + 0.438·21-s + 1.30·23-s + 0.200·25-s − 0.767·27-s − 0.802·29-s + 0.424·31-s − 1.38·33-s + 0.124·35-s + 0.676·37-s − 0.863·39-s − 0.142·41-s + 1.71·43-s − 0.664·45-s + 0.344·47-s − 0.922·49-s − 1.90·51-s + 0.933·53-s − 0.392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.274219457\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.274219457\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 0.736T + 7T^{2} \) |
| 11 | \( 1 - 2.91T + 11T^{2} \) |
| 13 | \( 1 - 1.97T + 13T^{2} \) |
| 17 | \( 1 - 4.98T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 - 6.25T + 23T^{2} \) |
| 29 | \( 1 + 4.32T + 29T^{2} \) |
| 31 | \( 1 - 2.36T + 31T^{2} \) |
| 37 | \( 1 - 4.11T + 37T^{2} \) |
| 41 | \( 1 + 0.914T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 2.35T + 47T^{2} \) |
| 53 | \( 1 - 6.79T + 53T^{2} \) |
| 59 | \( 1 - 1.43T + 59T^{2} \) |
| 61 | \( 1 + 6.61T + 61T^{2} \) |
| 67 | \( 1 + 0.618T + 67T^{2} \) |
| 71 | \( 1 + 1.73T + 71T^{2} \) |
| 73 | \( 1 + 2.26T + 73T^{2} \) |
| 79 | \( 1 - 0.939T + 79T^{2} \) |
| 83 | \( 1 + 0.163T + 83T^{2} \) |
| 89 | \( 1 + 0.156T + 89T^{2} \) |
| 97 | \( 1 - 2.90T + 97T^{2} \) |
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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
−7.77610267094330411485388199359, −7.26931692508774275622426942157, −6.56944096637559877673022778286, −5.84854802109253490209926628091, −5.39074854168549782063969482630, −4.57442690446511792659917758353, −3.75414859133430270147846295835, −2.99451069859149134412921825778, −1.28480568468355410586057962471, −0.77168841468397442370864762999,
0.77168841468397442370864762999, 1.28480568468355410586057962471, 2.99451069859149134412921825778, 3.75414859133430270147846295835, 4.57442690446511792659917758353, 5.39074854168549782063969482630, 5.84854802109253490209926628091, 6.56944096637559877673022778286, 7.26931692508774275622426942157, 7.77610267094330411485388199359