| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s + 5·7-s − 3·8-s − 2·9-s + 10-s − 2·11-s − 12-s + 2·13-s + 5·14-s + 15-s − 16-s − 2·18-s + 6·19-s − 20-s + 5·21-s − 2·22-s + 9·23-s − 3·24-s + 25-s + 2·26-s − 5·27-s − 5·28-s + 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.88·7-s − 1.06·8-s − 2/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.554·13-s + 1.33·14-s + 0.258·15-s − 1/4·16-s − 0.471·18-s + 1.37·19-s − 0.223·20-s + 1.09·21-s − 0.426·22-s + 1.87·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s − 0.962·27-s − 0.944·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.109101574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.109101574\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−9.154525972743801691967542274304, −8.803193927712396030361859047273, −8.043406735512049472154150448400, −7.25873984797176349142798903416, −5.79378556184675887096953301669, −5.23359085981744042930249949446, −4.66946282677540953692984388245, −3.44255909433744638508835564382, −2.60832335815397996539008924955, −1.25401043269288922318258428917,
1.25401043269288922318258428917, 2.60832335815397996539008924955, 3.44255909433744638508835564382, 4.66946282677540953692984388245, 5.23359085981744042930249949446, 5.79378556184675887096953301669, 7.25873984797176349142798903416, 8.043406735512049472154150448400, 8.803193927712396030361859047273, 9.154525972743801691967542274304
