L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s − 4·11-s + 13-s − 15-s − 17-s − 4·19-s − 4·21-s − 8·23-s + 25-s + 27-s − 6·29-s − 8·31-s − 4·33-s + 4·35-s − 2·37-s + 39-s + 6·41-s − 4·43-s − 45-s + 9·49-s − 51-s − 6·53-s + 4·55-s − 4·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.258·15-s − 0.242·17-s − 0.917·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.696·33-s + 0.676·35-s − 0.328·37-s + 0.160·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 9/7·49-s − 0.140·51-s − 0.824·53-s + 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.91817405151766, −15.32449881403410, −14.89052007862327, −14.19930800378621, −13.57229780165767, −13.12238855216221, −12.68539512638270, −12.36130777714039, −11.52102523906825, −10.74582184202266, −10.47559599699639, −9.789553526812361, −9.251002529378206, −8.806494640918883, −7.965044876210844, −7.684283749596563, −7.004399722841802, −6.256393800754498, −5.884131316727082, −5.043323484247466, −4.142779950152997, −3.713488777856412, −3.077670002545477, −2.408424288327939, −1.684884406822399, 0, 0,
1.684884406822399, 2.408424288327939, 3.077670002545477, 3.713488777856412, 4.142779950152997, 5.043323484247466, 5.884131316727082, 6.256393800754498, 7.004399722841802, 7.684283749596563, 7.965044876210844, 8.806494640918883, 9.251002529378206, 9.789553526812361, 10.47559599699639, 10.74582184202266, 11.52102523906825, 12.36130777714039, 12.68539512638270, 13.12238855216221, 13.57229780165767, 14.19930800378621, 14.89052007862327, 15.32449881403410, 15.91817405151766