L(s) = 1 | − 3-s − 2·5-s + 9-s + 4·11-s − 2·13-s + 2·15-s + 6·17-s − 4·19-s − 25-s − 27-s + 2·29-s − 4·33-s − 6·37-s + 2·39-s − 2·41-s − 4·43-s − 2·45-s − 6·51-s − 6·53-s − 8·55-s + 4·57-s − 12·59-s − 2·61-s + 4·65-s + 4·67-s + 6·73-s + 75-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s − 0.298·45-s − 0.840·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s − 1.56·59-s − 0.256·61-s + 0.496·65-s + 0.488·67-s + 0.702·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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
−7.44376428871628094569671391724, −6.55597781799330496824615300000, −6.19439119239525436838633052918, −5.16773217950502102961008704567, −4.64274308636669756274291953837, −3.75545050483136750219295120519, −3.35852253915733761610986992002, −2.03425588439014896949448393282, −1.08543961772187534196881977330, 0,
1.08543961772187534196881977330, 2.03425588439014896949448393282, 3.35852253915733761610986992002, 3.75545050483136750219295120519, 4.64274308636669756274291953837, 5.16773217950502102961008704567, 6.19439119239525436838633052918, 6.55597781799330496824615300000, 7.44376428871628094569671391724