L(s) = 1 | − 3.06·2-s + 3·3-s + 1.41·4-s − 10.3·5-s − 9.20·6-s − 32.3·7-s + 20.2·8-s + 9·9-s + 31.7·10-s + 11·11-s + 4.23·12-s − 34.5·13-s + 99.3·14-s − 31.0·15-s − 73.3·16-s + 42.1·17-s − 27.6·18-s − 107.·19-s − 14.6·20-s − 97.1·21-s − 33.7·22-s + 200.·23-s + 60.6·24-s − 18.1·25-s + 105.·26-s + 27·27-s − 45.7·28-s + ⋯ |
L(s) = 1 | − 1.08·2-s + 0.577·3-s + 0.176·4-s − 0.924·5-s − 0.626·6-s − 1.74·7-s + 0.893·8-s + 0.333·9-s + 1.00·10-s + 0.301·11-s + 0.101·12-s − 0.736·13-s + 1.89·14-s − 0.533·15-s − 1.14·16-s + 0.601·17-s − 0.361·18-s − 1.30·19-s − 0.163·20-s − 1.00·21-s − 0.327·22-s + 1.81·23-s + 0.515·24-s − 0.144·25-s + 0.798·26-s + 0.192·27-s − 0.308·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
good | 2 | \( 1 + 3.06T + 8T^{2} \) |
| 5 | \( 1 + 10.3T + 125T^{2} \) |
| 7 | \( 1 + 32.3T + 343T^{2} \) |
| 13 | \( 1 + 34.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 42.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 200.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 183.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 216.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 58.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 79.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 74.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 77.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 369.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 638.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 418.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 902.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 153.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 435.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 395.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 452.T + 9.12e5T^{2} \) |
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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
−8.571658841008048868752792979783, −7.74379495627256352576985871208, −7.05616042265005999979432224711, −6.51380698087292807398217474754, −5.06729047718584725652052825126, −4.02608927919342330295623200444, −3.35874083826442947263036172894, −2.34153436552495021628580106389, −0.845921665750147216766212509709, 0,
0.845921665750147216766212509709, 2.34153436552495021628580106389, 3.35874083826442947263036172894, 4.02608927919342330295623200444, 5.06729047718584725652052825126, 6.51380698087292807398217474754, 7.05616042265005999979432224711, 7.74379495627256352576985871208, 8.571658841008048868752792979783