L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 6.39·5-s + 6·6-s − 27.7·7-s + 8·8-s + 9·9-s − 12.7·10-s − 21.9·11-s + 12·12-s − 14.9·13-s − 55.4·14-s − 19.1·15-s + 16·16-s − 7.09·17-s + 18·18-s − 25.6·19-s − 25.5·20-s − 83.1·21-s − 43.9·22-s − 177.·23-s + 24·24-s − 84.1·25-s − 29.8·26-s + 27·27-s − 110.·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.571·5-s + 0.408·6-s − 1.49·7-s + 0.353·8-s + 0.333·9-s − 0.404·10-s − 0.602·11-s + 0.288·12-s − 0.318·13-s − 1.05·14-s − 0.330·15-s + 0.250·16-s − 0.101·17-s + 0.235·18-s − 0.310·19-s − 0.285·20-s − 0.864·21-s − 0.425·22-s − 1.60·23-s + 0.204·24-s − 0.672·25-s − 0.225·26-s + 0.192·27-s − 0.748·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 59 | \( 1 - 59T \) |
good | 5 | \( 1 + 6.39T + 125T^{2} \) |
| 7 | \( 1 + 27.7T + 343T^{2} \) |
| 11 | \( 1 + 21.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 7.09T + 4.91e3T^{2} \) |
| 19 | \( 1 + 25.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 76.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 234.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 198.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 410.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 84.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 586.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 609.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 66.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 729.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 40.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 193.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 76.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 536.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 972.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 599.T + 9.12e5T^{2} \) |
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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
−10.40726245870704875559595708517, −9.870233400072842743784402405470, −8.617453306818629703377088569207, −7.61883911368111790076843351312, −6.70166969268097947378253937646, −5.69185106854829735864340003064, −4.22950515213359610547220923063, −3.40189163156350408354148911438, −2.32853261818182609881162613003, 0,
2.32853261818182609881162613003, 3.40189163156350408354148911438, 4.22950515213359610547220923063, 5.69185106854829735864340003064, 6.70166969268097947378253937646, 7.61883911368111790076843351312, 8.617453306818629703377088569207, 9.870233400072842743784402405470, 10.40726245870704875559595708517