| L(s) = 1 | + 2-s − 4-s − 5-s − 2·7-s − 3·8-s − 10-s − 2·11-s − 2·14-s − 16-s + 7·17-s + 6·19-s + 20-s − 2·22-s + 6·23-s − 4·25-s + 2·28-s + 29-s − 4·31-s + 5·32-s + 7·34-s + 2·35-s − 37-s + 6·38-s + 3·40-s + 9·41-s + 6·43-s + 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.755·7-s − 1.06·8-s − 0.316·10-s − 0.603·11-s − 0.534·14-s − 1/4·16-s + 1.69·17-s + 1.37·19-s + 0.223·20-s − 0.426·22-s + 1.25·23-s − 4/5·25-s + 0.377·28-s + 0.185·29-s − 0.718·31-s + 0.883·32-s + 1.20·34-s + 0.338·35-s − 0.164·37-s + 0.973·38-s + 0.474·40-s + 1.40·41-s + 0.914·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.547960456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.547960456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−9.558473892030481203985778352711, −8.735214620811791573472916819842, −7.71658128317635772044098375577, −7.14815973994257176177139611517, −5.72247827094712758835446236838, −5.51771431794038684899397428080, −4.35432160798577722940495387148, −3.43785795084203303360022599835, −2.87973753020785058341151880462, −0.797230033462306644168187515692,
0.797230033462306644168187515692, 2.87973753020785058341151880462, 3.43785795084203303360022599835, 4.35432160798577722940495387148, 5.51771431794038684899397428080, 5.72247827094712758835446236838, 7.14815973994257176177139611517, 7.71658128317635772044098375577, 8.735214620811791573472916819842, 9.558473892030481203985778352711
