| L(s) = 1 | + 0.468·2-s + 3-s − 1.78·4-s − 3.25·5-s + 0.468·6-s − 3.43·7-s − 1.77·8-s + 9-s − 1.52·10-s − 11-s − 1.78·12-s − 6.24·13-s − 1.61·14-s − 3.25·15-s + 2.72·16-s + 4.43·17-s + 0.468·18-s + 2.00·19-s + 5.79·20-s − 3.43·21-s − 0.468·22-s − 5.24·23-s − 1.77·24-s + 5.60·25-s − 2.92·26-s + 27-s + 6.11·28-s + ⋯ |
| L(s) = 1 | + 0.331·2-s + 0.577·3-s − 0.890·4-s − 1.45·5-s + 0.191·6-s − 1.29·7-s − 0.626·8-s + 0.333·9-s − 0.482·10-s − 0.301·11-s − 0.513·12-s − 1.73·13-s − 0.430·14-s − 0.840·15-s + 0.682·16-s + 1.07·17-s + 0.110·18-s + 0.459·19-s + 1.29·20-s − 0.749·21-s − 0.0999·22-s − 1.09·23-s − 0.361·24-s + 1.12·25-s − 0.574·26-s + 0.192·27-s + 1.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7115433700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7115433700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.468T + 2T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 + 3.43T + 7T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 - 4.43T + 17T^{2} \) |
| 19 | \( 1 - 2.00T + 19T^{2} \) |
| 23 | \( 1 + 5.24T + 23T^{2} \) |
| 29 | \( 1 - 2.01T + 29T^{2} \) |
| 31 | \( 1 - 2.78T + 31T^{2} \) |
| 37 | \( 1 - 9.46T + 37T^{2} \) |
| 41 | \( 1 + 6.90T + 41T^{2} \) |
| 43 | \( 1 - 7.06T + 43T^{2} \) |
| 47 | \( 1 + 3.28T + 47T^{2} \) |
| 53 | \( 1 + 3.39T + 53T^{2} \) |
| 59 | \( 1 - 6.95T + 59T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 1.02T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 5.93T + 89T^{2} \) |
| 97 | \( 1 - 1.30T + 97T^{2} \) |
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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
−9.283293299970366805466557286325, −8.128885083995254771684891307675, −7.81204900827850438831720746695, −6.99542205797344645770709887488, −5.88607207628392766744875342353, −4.85286533494802948610797105085, −4.14006816018649709275485047540, −3.35492948188325443892878369062, −2.75176150109953988056667660383, −0.49627797060063445364156980608,
0.49627797060063445364156980608, 2.75176150109953988056667660383, 3.35492948188325443892878369062, 4.14006816018649709275485047540, 4.85286533494802948610797105085, 5.88607207628392766744875342353, 6.99542205797344645770709887488, 7.81204900827850438831720746695, 8.128885083995254771684891307675, 9.283293299970366805466557286325
