| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s + 2·11-s − 12-s − 13-s − 14-s + 16-s − 17-s − 18-s − 5·19-s − 21-s − 2·22-s − 2·23-s + 24-s − 5·25-s + 26-s − 27-s + 28-s − 10·29-s + 31-s − 32-s − 2·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.14·19-s − 0.218·21-s − 0.426·22-s − 0.417·23-s + 0.204·24-s − 25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s − 1.85·29-s + 0.179·31-s − 0.176·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8114319535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8114319535\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.73621862687142673774590843378, −7.07970912053862759969918825083, −6.37424907924857194602218926786, −5.85138990676734306134415162538, −5.03285179559074951692054298740, −4.18699265359558771540354304646, −3.54793959421640908870748580437, −2.20209028121746292817743471832, −1.73342905192421108229963396270, −0.48905418956000582334471169738,
0.48905418956000582334471169738, 1.73342905192421108229963396270, 2.20209028121746292817743471832, 3.54793959421640908870748580437, 4.18699265359558771540354304646, 5.03285179559074951692054298740, 5.85138990676734306134415162538, 6.37424907924857194602218926786, 7.07970912053862759969918825083, 7.73621862687142673774590843378
