| L(s) = 1 | − 2i·3-s + i·7-s − 9-s − 5·11-s − 8i·17-s − 2·19-s + 2·21-s + 7i·23-s − 4i·27-s + 3·29-s − 4·31-s + 10i·33-s + i·37-s − 2·41-s − 3i·43-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + 0.377i·7-s − 0.333·9-s − 1.50·11-s − 1.94i·17-s − 0.458·19-s + 0.436·21-s + 1.45i·23-s − 0.769i·27-s + 0.557·29-s − 0.718·31-s + 1.74i·33-s + 0.164i·37-s − 0.312·41-s − 0.457i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 7iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 3iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−8.064499817756993566215652899860, −7.41331755143394449844783082373, −7.08094170044652930332358686339, −5.96027751751010371658031391778, −5.34752915786799963321742546700, −4.49333666902057687959281748280, −3.01712644156276181559076837877, −2.45718304305413496758433000770, −1.33611695416700141873696129895, 0,
1.84205090641577382071622873419, 2.99469315841616912960814690385, 3.90668781513470109811689282590, 4.54782058936440789788789985950, 5.28379601360898431998292633966, 6.15274325143092457654797980340, 7.01884629898722068219912796360, 8.133685211579967905936760095884, 8.413086864234618494873052695891
