L(s) = 1 | + 2.62·2-s + 3-s + 4.87·4-s + 2.54·5-s + 2.62·6-s + 7-s + 7.52·8-s + 9-s + 6.68·10-s + 2.77·11-s + 4.87·12-s − 5.75·13-s + 2.62·14-s + 2.54·15-s + 9.98·16-s − 2.01·17-s + 2.62·18-s + 0.371·19-s + 12.4·20-s + 21-s + 7.27·22-s − 0.539·23-s + 7.52·24-s + 1.50·25-s − 15.0·26-s + 27-s + 4.87·28-s + ⋯ |
L(s) = 1 | + 1.85·2-s + 0.577·3-s + 2.43·4-s + 1.14·5-s + 1.07·6-s + 0.377·7-s + 2.66·8-s + 0.333·9-s + 2.11·10-s + 0.836·11-s + 1.40·12-s − 1.59·13-s + 0.700·14-s + 0.658·15-s + 2.49·16-s − 0.489·17-s + 0.617·18-s + 0.0851·19-s + 2.77·20-s + 0.218·21-s + 1.55·22-s − 0.112·23-s + 1.53·24-s + 0.300·25-s − 2.95·26-s + 0.192·27-s + 0.920·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.702925077\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.702925077\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 191 | \( 1 - T \) |
good | 2 | \( 1 - 2.62T + 2T^{2} \) |
| 5 | \( 1 - 2.54T + 5T^{2} \) |
| 11 | \( 1 - 2.77T + 11T^{2} \) |
| 13 | \( 1 + 5.75T + 13T^{2} \) |
| 17 | \( 1 + 2.01T + 17T^{2} \) |
| 19 | \( 1 - 0.371T + 19T^{2} \) |
| 23 | \( 1 + 0.539T + 23T^{2} \) |
| 29 | \( 1 + 9.57T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 + 5.84T + 37T^{2} \) |
| 41 | \( 1 - 3.70T + 41T^{2} \) |
| 43 | \( 1 + 7.69T + 43T^{2} \) |
| 47 | \( 1 - 5.96T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 3.29T + 61T^{2} \) |
| 67 | \( 1 - 1.02T + 67T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 - 7.33T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 - 4.51T + 97T^{2} \) |
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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
−8.281993550383866088168692312908, −7.27916156256509531226149108647, −6.85192012225021937063287907939, −6.01151188988275983532302403160, −5.34687794727988380627972710964, −4.67992672840124204230878503396, −3.97283831830459002201855203602, −3.01879422343173941233866175447, −2.20213973389284890245155641658, −1.70649362404382946687640212640,
1.70649362404382946687640212640, 2.20213973389284890245155641658, 3.01879422343173941233866175447, 3.97283831830459002201855203602, 4.67992672840124204230878503396, 5.34687794727988380627972710964, 6.01151188988275983532302403160, 6.85192012225021937063287907939, 7.27916156256509531226149108647, 8.281993550383866088168692312908