L(s) = 1 | − 2-s − 3.21·3-s + 4-s + 5-s + 3.21·6-s − 0.0233·7-s − 8-s + 7.34·9-s − 10-s + 2.16·11-s − 3.21·12-s − 1.59·13-s + 0.0233·14-s − 3.21·15-s + 16-s − 7.97·17-s − 7.34·18-s + 20-s + 0.0752·21-s − 2.16·22-s + 4.48·23-s + 3.21·24-s + 25-s + 1.59·26-s − 13.9·27-s − 0.0233·28-s + 7.55·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.85·3-s + 0.5·4-s + 0.447·5-s + 1.31·6-s − 0.00884·7-s − 0.353·8-s + 2.44·9-s − 0.316·10-s + 0.651·11-s − 0.928·12-s − 0.441·13-s + 0.00625·14-s − 0.830·15-s + 0.250·16-s − 1.93·17-s − 1.73·18-s + 0.223·20-s + 0.0164·21-s − 0.460·22-s + 0.934·23-s + 0.656·24-s + 0.200·25-s + 0.311·26-s − 2.68·27-s − 0.00442·28-s + 1.40·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6497228328\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6497228328\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 3.21T + 3T^{2} \) |
| 7 | \( 1 + 0.0233T + 7T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 13 | \( 1 + 1.59T + 13T^{2} \) |
| 17 | \( 1 + 7.97T + 17T^{2} \) |
| 23 | \( 1 - 4.48T + 23T^{2} \) |
| 29 | \( 1 - 7.55T + 29T^{2} \) |
| 31 | \( 1 - 4.45T + 31T^{2} \) |
| 37 | \( 1 + 0.389T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 - 9.70T + 43T^{2} \) |
| 47 | \( 1 - 4.27T + 47T^{2} \) |
| 53 | \( 1 + 3.78T + 53T^{2} \) |
| 59 | \( 1 + 3.63T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 3.85T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 1.90T + 73T^{2} \) |
| 79 | \( 1 + 9.51T + 79T^{2} \) |
| 83 | \( 1 + 2.62T + 83T^{2} \) |
| 89 | \( 1 - 8.87T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−8.785075672086052597988941373334, −7.56607775542748753108879899436, −6.83743204378689136184106057695, −6.38707427094561250274252081997, −5.82444187120280329955987095101, −4.71694576562605715915701254410, −4.39730291572183119394044561317, −2.71475510259387992665019541360, −1.55864457226251349293745109696, −0.60587022598474903503873201839,
0.60587022598474903503873201839, 1.55864457226251349293745109696, 2.71475510259387992665019541360, 4.39730291572183119394044561317, 4.71694576562605715915701254410, 5.82444187120280329955987095101, 6.38707427094561250274252081997, 6.83743204378689136184106057695, 7.56607775542748753108879899436, 8.785075672086052597988941373334