| L(s) = 1 | + 2-s − 3.22·3-s + 4-s + 3.63·5-s − 3.22·6-s − 0.166·7-s + 8-s + 7.38·9-s + 3.63·10-s + 2.07·11-s − 3.22·12-s + 5.70·13-s − 0.166·14-s − 11.7·15-s + 16-s − 7.23·17-s + 7.38·18-s − 19-s + 3.63·20-s + 0.536·21-s + 2.07·22-s − 0.228·23-s − 3.22·24-s + 8.22·25-s + 5.70·26-s − 14.1·27-s − 0.166·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.86·3-s + 0.5·4-s + 1.62·5-s − 1.31·6-s − 0.0628·7-s + 0.353·8-s + 2.46·9-s + 1.14·10-s + 0.624·11-s − 0.930·12-s + 1.58·13-s − 0.0444·14-s − 3.02·15-s + 0.250·16-s − 1.75·17-s + 1.74·18-s − 0.229·19-s + 0.813·20-s + 0.116·21-s + 0.441·22-s − 0.0476·23-s − 0.657·24-s + 1.64·25-s + 1.11·26-s − 2.71·27-s − 0.0314·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.844005307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.844005307\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 3.22T + 3T^{2} \) |
| 5 | \( 1 - 3.63T + 5T^{2} \) |
| 7 | \( 1 + 0.166T + 7T^{2} \) |
| 11 | \( 1 - 2.07T + 11T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 17 | \( 1 + 7.23T + 17T^{2} \) |
| 23 | \( 1 + 0.228T + 23T^{2} \) |
| 29 | \( 1 - 5.83T + 29T^{2} \) |
| 31 | \( 1 + 6.23T + 31T^{2} \) |
| 37 | \( 1 - 5.72T + 37T^{2} \) |
| 41 | \( 1 - 7.52T + 41T^{2} \) |
| 43 | \( 1 - 6.52T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 7.72T + 53T^{2} \) |
| 59 | \( 1 - 0.482T + 59T^{2} \) |
| 61 | \( 1 - 6.01T + 61T^{2} \) |
| 67 | \( 1 - 3.78T + 67T^{2} \) |
| 71 | \( 1 - 9.35T + 71T^{2} \) |
| 73 | \( 1 + 5.89T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 8.79T + 83T^{2} \) |
| 89 | \( 1 - 18.4T + 89T^{2} \) |
| 97 | \( 1 + 0.277T + 97T^{2} \) |
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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
−7.35228185025765071279497030745, −6.56597331362147679474288028419, −6.23359276171613614671665561931, −5.93912909469772681768833714250, −5.22478796233950103350483034696, −4.44851371133037657572186301747, −3.93157511816123902954346780641, −2.48571069424920845838437014236, −1.64083668811468159606213095225, −0.889589850915807711401421635251,
0.889589850915807711401421635251, 1.64083668811468159606213095225, 2.48571069424920845838437014236, 3.93157511816123902954346780641, 4.44851371133037657572186301747, 5.22478796233950103350483034696, 5.93912909469772681768833714250, 6.23359276171613614671665561931, 6.56597331362147679474288028419, 7.35228185025765071279497030745
