| L(s) = 1 | − 3-s − 2·5-s + 9-s − 4·11-s − 13-s + 2·15-s − 2·17-s − 25-s − 27-s − 10·29-s + 4·31-s + 4·33-s − 2·37-s + 39-s − 6·41-s + 12·43-s − 2·45-s + 2·51-s + 6·53-s + 8·55-s + 12·59-s + 2·61-s + 2·65-s + 8·67-s − 2·73-s + 75-s − 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.516·15-s − 0.485·17-s − 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s + 0.696·33-s − 0.328·37-s + 0.160·39-s − 0.937·41-s + 1.82·43-s − 0.298·45-s + 0.280·51-s + 0.824·53-s + 1.07·55-s + 1.56·59-s + 0.256·61-s + 0.248·65-s + 0.977·67-s − 0.234·73-s + 0.115·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.53010466803154, −14.93664341761795, −14.50846183855011, −13.47401739448955, −13.37809250317037, −12.57325709945134, −12.27265827230249, −11.43654309980905, −11.31810807888510, −10.61553328831865, −10.12948796583513, −9.531804049543343, −8.803544581174205, −8.189497811244830, −7.662522011279366, −7.203777720384798, −6.651329887923801, −5.663418770332203, −5.465199011426262, −4.641061188336648, −4.075561897083643, −3.478435357185357, −2.564290163419404, −1.944216926363790, −0.7176942661661051, 0,
0.7176942661661051, 1.944216926363790, 2.564290163419404, 3.478435357185357, 4.075561897083643, 4.641061188336648, 5.465199011426262, 5.663418770332203, 6.651329887923801, 7.203777720384798, 7.662522011279366, 8.189497811244830, 8.803544581174205, 9.531804049543343, 10.12948796583513, 10.61553328831865, 11.31810807888510, 11.43654309980905, 12.27265827230249, 12.57325709945134, 13.37809250317037, 13.47401739448955, 14.50846183855011, 14.93664341761795, 15.53010466803154
