| L(s) = 1 | + 2·2-s + 5·3-s − 4·4-s + 10·6-s + 8·7-s − 24·8-s − 2·9-s + 34·11-s − 20·12-s + 57·13-s + 16·14-s − 16·16-s + 80·17-s − 4·18-s − 70·19-s + 40·21-s + 68·22-s − 23·23-s − 120·24-s + 114·26-s − 145·27-s − 32·28-s + 245·29-s + 103·31-s + 160·32-s + 170·33-s + 160·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.962·3-s − 1/2·4-s + 0.680·6-s + 0.431·7-s − 1.06·8-s − 0.0740·9-s + 0.931·11-s − 0.481·12-s + 1.21·13-s + 0.305·14-s − 1/4·16-s + 1.14·17-s − 0.0523·18-s − 0.845·19-s + 0.415·21-s + 0.658·22-s − 0.208·23-s − 1.02·24-s + 0.859·26-s − 1.03·27-s − 0.215·28-s + 1.56·29-s + 0.596·31-s + 0.883·32-s + 0.896·33-s + 0.807·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.740283438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.740283438\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + p T \) |
| good | 2 | \( 1 - p T + p^{3} T^{2} \) |
| 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 34 T + p^{3} T^{2} \) |
| 13 | \( 1 - 57 T + p^{3} T^{2} \) |
| 17 | \( 1 - 80 T + p^{3} T^{2} \) |
| 19 | \( 1 + 70 T + p^{3} T^{2} \) |
| 29 | \( 1 - 245 T + p^{3} T^{2} \) |
| 31 | \( 1 - 103 T + p^{3} T^{2} \) |
| 37 | \( 1 - 298 T + p^{3} T^{2} \) |
| 41 | \( 1 - 95 T + p^{3} T^{2} \) |
| 43 | \( 1 + 88 T + p^{3} T^{2} \) |
| 47 | \( 1 - 357 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 408 T + p^{3} T^{2} \) |
| 61 | \( 1 - 822 T + p^{3} T^{2} \) |
| 67 | \( 1 + 926 T + p^{3} T^{2} \) |
| 71 | \( 1 - 335 T + p^{3} T^{2} \) |
| 73 | \( 1 - 899 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1322 T + p^{3} T^{2} \) |
| 83 | \( 1 - 36 T + p^{3} T^{2} \) |
| 89 | \( 1 + 460 T + p^{3} T^{2} \) |
| 97 | \( 1 - 964 T + p^{3} T^{2} \) |
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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
−10.18878259711087522987117644625, −9.222230238644548632528482683768, −8.533611235985534706587727285801, −8.000477942233750832795502106264, −6.46344133549010505109306422705, −5.68532995017926889763036760381, −4.39199625237408202210970535066, −3.69600526585573277192420485864, −2.70520533408397225556818261618, −1.08832076135608081015582906932,
1.08832076135608081015582906932, 2.70520533408397225556818261618, 3.69600526585573277192420485864, 4.39199625237408202210970535066, 5.68532995017926889763036760381, 6.46344133549010505109306422705, 8.000477942233750832795502106264, 8.533611235985534706587727285801, 9.222230238644548632528482683768, 10.18878259711087522987117644625
