L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s + 3·7-s + 9-s − 2·12-s − 13-s − 6·14-s − 4·16-s − 2·17-s − 2·18-s + 5·19-s − 3·21-s + 6·23-s + 2·26-s − 27-s + 6·28-s − 10·29-s − 3·31-s + 8·32-s + 4·34-s + 2·36-s + 2·37-s − 10·38-s + 39-s + 8·41-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 1.13·7-s + 1/3·9-s − 0.577·12-s − 0.277·13-s − 1.60·14-s − 16-s − 0.485·17-s − 0.471·18-s + 1.14·19-s − 0.654·21-s + 1.25·23-s + 0.392·26-s − 0.192·27-s + 1.13·28-s − 1.85·29-s − 0.538·31-s + 1.41·32-s + 0.685·34-s + 1/3·36-s + 0.328·37-s − 1.62·38-s + 0.160·39-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.63445690246110755414862928105, −7.06328550761489343142245284993, −6.16165631359848537381830780833, −5.26550699194046601552455316284, −4.80645524398487230267862763673, −3.93276270400122129280671872839, −2.70866001385341573464253183059, −1.71419845103506630043143205825, −1.13781757996508329611151326735, 0,
1.13781757996508329611151326735, 1.71419845103506630043143205825, 2.70866001385341573464253183059, 3.93276270400122129280671872839, 4.80645524398487230267862763673, 5.26550699194046601552455316284, 6.16165631359848537381830780833, 7.06328550761489343142245284993, 7.63445690246110755414862928105