| L(s) = 1 | − 2i·3-s − 2i·7-s − 9-s − 4·11-s + 6i·13-s − 2i·17-s − 8·19-s − 4·21-s + 6i·23-s − 4i·27-s − 2·29-s − 4·31-s + 8i·33-s + 2i·37-s + 12·39-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s − 0.755i·7-s − 0.333·9-s − 1.20·11-s + 1.66i·13-s − 0.485i·17-s − 1.83·19-s − 0.872·21-s + 1.25i·23-s − 0.769i·27-s − 0.371·29-s − 0.718·31-s + 1.39i·33-s + 0.328i·37-s + 1.92·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 6iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.680928383242480127954456467440, −7.947159410057509323919136631618, −7.03480239937336302191128047163, −6.86375252968661253189757882435, −5.73752957257773745987183120060, −4.65346565249472860870920095398, −3.77112147073722392183692579348, −2.32443200779169547829008970144, −1.59187482730967088947058848206, 0,
2.21913351617222597891132330615, 3.12607651914747200329058045298, 4.13537087465230804571820996375, 5.06632952221844786878583681668, 5.60104124022750292102211641871, 6.56907834446500933513654921958, 7.87828404512942150249527675303, 8.409799530723711511241889912968, 9.155985583096117039394650441908
