| L(s) = 1 | − 2.54·2-s − 0.805·3-s + 4.50·4-s − 2.33·5-s + 2.05·6-s + 2.59·7-s − 6.37·8-s − 2.35·9-s + 5.95·10-s − 5.57·11-s − 3.62·12-s + 4.69·13-s − 6.60·14-s + 1.88·15-s + 7.26·16-s − 5.12·17-s + 5.99·18-s − 1.42·19-s − 10.5·20-s − 2.08·21-s + 14.2·22-s + 23-s + 5.13·24-s + 0.459·25-s − 11.9·26-s + 4.30·27-s + 11.6·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 0.464·3-s + 2.25·4-s − 1.04·5-s + 0.838·6-s + 0.979·7-s − 2.25·8-s − 0.783·9-s + 1.88·10-s − 1.68·11-s − 1.04·12-s + 1.30·13-s − 1.76·14-s + 0.485·15-s + 1.81·16-s − 1.24·17-s + 1.41·18-s − 0.327·19-s − 2.35·20-s − 0.455·21-s + 3.02·22-s + 0.208·23-s + 1.04·24-s + 0.0919·25-s − 2.34·26-s + 0.829·27-s + 2.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2016675265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2016675265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 349 | \( 1 + T \) |
| good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 3 | \( 1 + 0.805T + 3T^{2} \) |
| 5 | \( 1 + 2.33T + 5T^{2} \) |
| 7 | \( 1 - 2.59T + 7T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 - 4.69T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 + 1.42T + 19T^{2} \) |
| 29 | \( 1 - 6.55T + 29T^{2} \) |
| 31 | \( 1 - 3.56T + 31T^{2} \) |
| 37 | \( 1 + 5.48T + 37T^{2} \) |
| 41 | \( 1 + 0.275T + 41T^{2} \) |
| 43 | \( 1 + 3.49T + 43T^{2} \) |
| 47 | \( 1 + 2.00T + 47T^{2} \) |
| 53 | \( 1 + 1.80T + 53T^{2} \) |
| 59 | \( 1 - 2.39T + 59T^{2} \) |
| 61 | \( 1 + 4.11T + 61T^{2} \) |
| 67 | \( 1 + 2.86T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 4.26T + 79T^{2} \) |
| 83 | \( 1 + 8.54T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.169647134192579849613498593072, −7.43772936897717184123541342685, −6.71023101058255940442676421230, −6.02955201803423291699762455940, −5.12273131019771250824737789619, −4.38216298097671024364069862962, −3.13788215440788804699955440970, −2.42156730016996662758591529168, −1.40213864077940077583463571873, −0.31127312226563963506987359884,
0.31127312226563963506987359884, 1.40213864077940077583463571873, 2.42156730016996662758591529168, 3.13788215440788804699955440970, 4.38216298097671024364069862962, 5.12273131019771250824737789619, 6.02955201803423291699762455940, 6.71023101058255940442676421230, 7.43772936897717184123541342685, 8.169647134192579849613498593072
