L(s) = 1 | + 3.80·2-s − 6.79·3-s + 6.47·4-s + 10.6·5-s − 25.8·6-s + 11.3·7-s − 5.79·8-s + 19.1·9-s + 40.6·10-s − 20.8·11-s − 44.0·12-s − 39.2·13-s + 43.0·14-s − 72.6·15-s − 73.8·16-s + 32.2·17-s + 72.8·18-s + 10.1·19-s + 69.2·20-s − 76.9·21-s − 79.3·22-s + 23·23-s + 39.3·24-s − 10.6·25-s − 149.·26-s + 53.3·27-s + 73.3·28-s + ⋯ |
L(s) = 1 | + 1.34·2-s − 1.30·3-s + 0.809·4-s + 0.956·5-s − 1.75·6-s + 0.611·7-s − 0.256·8-s + 0.709·9-s + 1.28·10-s − 0.571·11-s − 1.05·12-s − 0.838·13-s + 0.822·14-s − 1.25·15-s − 1.15·16-s + 0.460·17-s + 0.954·18-s + 0.122·19-s + 0.774·20-s − 0.799·21-s − 0.769·22-s + 0.208·23-s + 0.334·24-s − 0.0853·25-s − 1.12·26-s + 0.379·27-s + 0.495·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
good | 2 | \( 1 - 3.80T + 8T^{2} \) |
| 3 | \( 1 + 6.79T + 27T^{2} \) |
| 5 | \( 1 - 10.6T + 125T^{2} \) |
| 7 | \( 1 - 11.3T + 343T^{2} \) |
| 11 | \( 1 + 20.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 39.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 32.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 10.1T + 6.85e3T^{2} \) |
| 31 | \( 1 + 82.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 219.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 164.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 106.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 35.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 336.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 834.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 88.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 978.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 476.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.05e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 683.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 768.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 12.1T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3T + 9.12e5T^{2} \) |
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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
−10.03983989150733385301126792593, −8.954462349222613615835076409169, −7.53438754993043857077872962331, −6.50996262491640330410313502195, −5.68529980511897047792532366801, −5.21991138467807229391141635912, −4.55482430479369974762393185789, −3.06219057606474131895101181044, −1.78343731860665684562565802673, 0,
1.78343731860665684562565802673, 3.06219057606474131895101181044, 4.55482430479369974762393185789, 5.21991138467807229391141635912, 5.68529980511897047792532366801, 6.50996262491640330410313502195, 7.53438754993043857077872962331, 8.954462349222613615835076409169, 10.03983989150733385301126792593