| L(s) = 1 | + 1.53·2-s + 3-s + 0.364·4-s − 2.97·5-s + 1.53·6-s + 1.52·7-s − 2.51·8-s + 9-s − 4.56·10-s + 0.657·11-s + 0.364·12-s + 3.12·13-s + 2.34·14-s − 2.97·15-s − 4.59·16-s + 1.99·17-s + 1.53·18-s + 6.50·19-s − 1.08·20-s + 1.52·21-s + 1.01·22-s − 23-s − 2.51·24-s + 3.82·25-s + 4.80·26-s + 27-s + 0.554·28-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.577·3-s + 0.182·4-s − 1.32·5-s + 0.627·6-s + 0.575·7-s − 0.889·8-s + 0.333·9-s − 1.44·10-s + 0.198·11-s + 0.105·12-s + 0.866·13-s + 0.625·14-s − 0.767·15-s − 1.14·16-s + 0.483·17-s + 0.362·18-s + 1.49·19-s − 0.241·20-s + 0.332·21-s + 0.215·22-s − 0.208·23-s − 0.513·24-s + 0.765·25-s + 0.942·26-s + 0.192·27-s + 0.104·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.025571729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.025571729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 5 | \( 1 + 2.97T + 5T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 - 0.657T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 - 1.99T + 17T^{2} \) |
| 19 | \( 1 - 6.50T + 19T^{2} \) |
| 31 | \( 1 - 6.58T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 - 8.68T + 41T^{2} \) |
| 43 | \( 1 + 3.14T + 43T^{2} \) |
| 47 | \( 1 - 1.88T + 47T^{2} \) |
| 53 | \( 1 + 2.78T + 53T^{2} \) |
| 59 | \( 1 - 6.26T + 59T^{2} \) |
| 61 | \( 1 - 1.61T + 61T^{2} \) |
| 67 | \( 1 - 5.54T + 67T^{2} \) |
| 71 | \( 1 + 9.71T + 71T^{2} \) |
| 73 | \( 1 - 4.17T + 73T^{2} \) |
| 79 | \( 1 - 9.99T + 79T^{2} \) |
| 83 | \( 1 - 3.81T + 83T^{2} \) |
| 89 | \( 1 + 0.899T + 89T^{2} \) |
| 97 | \( 1 + 3.81T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.062855729434628473453739106843, −8.085072159835518155350345370410, −7.84123404687167074862373705977, −6.75708607474593179464911600621, −5.78271643589618309245987088459, −4.87451216660592673508097115056, −4.11002880687227800878465414028, −3.56139603733601333710676285640, −2.74187615401034879752643284658, −1.01682846924476229434422161317,
1.01682846924476229434422161317, 2.74187615401034879752643284658, 3.56139603733601333710676285640, 4.11002880687227800878465414028, 4.87451216660592673508097115056, 5.78271643589618309245987088459, 6.75708607474593179464911600621, 7.84123404687167074862373705977, 8.085072159835518155350345370410, 9.062855729434628473453739106843
