| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 11-s − 12-s + 1.41·13-s + 16-s + 4·17-s − 18-s + 5.41·19-s − 22-s + 7.65·23-s + 24-s − 5·25-s − 1.41·26-s − 27-s − 5.65·29-s − 3.07·31-s − 32-s − 33-s − 4·34-s + 36-s + 3.65·37-s − 5.41·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.301·11-s − 0.288·12-s + 0.392·13-s + 0.250·16-s + 0.970·17-s − 0.235·18-s + 1.24·19-s − 0.213·22-s + 1.59·23-s + 0.204·24-s − 25-s − 0.277·26-s − 0.192·27-s − 1.05·29-s − 0.551·31-s − 0.176·32-s − 0.174·33-s − 0.685·34-s + 0.166·36-s + 0.601·37-s − 0.878·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.174241190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.174241190\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 5.41T + 19T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 + 3.07T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 - 9.65T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.766960427947677118543865671761, −7.72148591126973706192010012839, −7.37667243631191563309784703290, −6.44899630446655505574090735130, −5.68576150066804518200048010910, −5.06334681038250510709292966591, −3.82779244373936436383246083855, −3.04439911837506710661099257336, −1.68477810551715713715733220735, −0.78709164801638357644427046107,
0.78709164801638357644427046107, 1.68477810551715713715733220735, 3.04439911837506710661099257336, 3.82779244373936436383246083855, 5.06334681038250510709292966591, 5.68576150066804518200048010910, 6.44899630446655505574090735130, 7.37667243631191563309784703290, 7.72148591126973706192010012839, 8.766960427947677118543865671761
