L(s) = 1 | + 1.90·2-s + 3-s + 1.62·4-s − 2.90·5-s + 1.90·6-s − 7-s − 0.719·8-s + 9-s − 5.52·10-s + 3.52·11-s + 1.62·12-s + 6.90·13-s − 1.90·14-s − 2.90·15-s − 4.61·16-s − 3.37·17-s + 1.90·18-s + 1.24·19-s − 4.70·20-s − 21-s + 6.70·22-s − 6.23·23-s − 0.719·24-s + 3.42·25-s + 13.1·26-s + 27-s − 1.62·28-s + ⋯ |
L(s) = 1 | + 1.34·2-s + 0.577·3-s + 0.811·4-s − 1.29·5-s + 0.776·6-s − 0.377·7-s − 0.254·8-s + 0.333·9-s − 1.74·10-s + 1.06·11-s + 0.468·12-s + 1.91·13-s − 0.508·14-s − 0.749·15-s − 1.15·16-s − 0.819·17-s + 0.448·18-s + 0.285·19-s − 1.05·20-s − 0.218·21-s + 1.43·22-s − 1.30·23-s − 0.146·24-s + 0.685·25-s + 2.57·26-s + 0.192·27-s − 0.306·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.038659625\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.038659625\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 5 | \( 1 + 2.90T + 5T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 13 | \( 1 - 6.90T + 13T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 - 7.67T + 29T^{2} \) |
| 31 | \( 1 + 6.62T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 1.95T + 41T^{2} \) |
| 43 | \( 1 + 1.80T + 43T^{2} \) |
| 47 | \( 1 - 7.05T + 47T^{2} \) |
| 53 | \( 1 - 4.62T + 53T^{2} \) |
| 59 | \( 1 - 5.24T + 59T^{2} \) |
| 61 | \( 1 - 3.76T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 0.133T + 71T^{2} \) |
| 73 | \( 1 - 9.61T + 73T^{2} \) |
| 79 | \( 1 + 6.28T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 1.65T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−7.76173217053734405084808961330, −6.95626880866091788389249530946, −6.28875515635303599691992063827, −5.87458893263107382238119635583, −4.64310176018725541421545408015, −4.04969877531892462725578588543, −3.74908395668945466475011009630, −3.17113184285583068566504932057, −2.09760984529764764506328104204, −0.789895016493509473108768913838,
0.789895016493509473108768913838, 2.09760984529764764506328104204, 3.17113184285583068566504932057, 3.74908395668945466475011009630, 4.04969877531892462725578588543, 4.64310176018725541421545408015, 5.87458893263107382238119635583, 6.28875515635303599691992063827, 6.95626880866091788389249530946, 7.76173217053734405084808961330