L(s) = 1 | + 2.45·2-s + 4.04·4-s + 2.05·5-s + 7-s + 5.01·8-s + 5.04·10-s + 0.206·13-s + 2.45·14-s + 4.24·16-s + 0.813·17-s + 6.28·19-s + 8.28·20-s − 5.11·23-s − 0.793·25-s + 0.507·26-s + 4.04·28-s − 3.87·29-s + 6.49·31-s + 0.406·32-s + 2·34-s + 2.05·35-s + 1.79·37-s + 15.4·38-s + 10.2·40-s − 1.82·41-s + 10.0·43-s − 12.5·46-s + ⋯ |
L(s) = 1 | + 1.73·2-s + 2.02·4-s + 0.917·5-s + 0.377·7-s + 1.77·8-s + 1.59·10-s + 0.0572·13-s + 0.656·14-s + 1.06·16-s + 0.197·17-s + 1.44·19-s + 1.85·20-s − 1.06·23-s − 0.158·25-s + 0.0995·26-s + 0.763·28-s − 0.720·29-s + 1.16·31-s + 0.0719·32-s + 0.342·34-s + 0.346·35-s + 0.294·37-s + 2.50·38-s + 1.62·40-s − 0.285·41-s + 1.53·43-s − 1.85·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.243792789\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.243792789\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 5 | \( 1 - 2.05T + 5T^{2} \) |
| 13 | \( 1 - 0.206T + 13T^{2} \) |
| 17 | \( 1 - 0.813T + 17T^{2} \) |
| 19 | \( 1 - 6.28T + 19T^{2} \) |
| 23 | \( 1 + 5.11T + 23T^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 - 6.49T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 8.79T + 47T^{2} \) |
| 53 | \( 1 - 3.08T + 53T^{2} \) |
| 59 | \( 1 + 2.66T + 59T^{2} \) |
| 61 | \( 1 - 3.58T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 7.79T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 - 5.72T + 89T^{2} \) |
| 97 | \( 1 + 8.49T + 97T^{2} \) |
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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
−7.60240991870920571468458590888, −6.93032170693981902900003310673, −6.15094407306317303843525059974, −5.57174868293700045040529786541, −5.26833682315082505283583462297, −4.27088801166915891462043122279, −3.75595522709663940746520290204, −2.74228934061646963522479251641, −2.19879209197845954973301645805, −1.19886881378098332071349691802,
1.19886881378098332071349691802, 2.19879209197845954973301645805, 2.74228934061646963522479251641, 3.75595522709663940746520290204, 4.27088801166915891462043122279, 5.26833682315082505283583462297, 5.57174868293700045040529786541, 6.15094407306317303843525059974, 6.93032170693981902900003310673, 7.60240991870920571468458590888