| L(s) = 1 | + (1 + i)2-s − 3-s + 2i·4-s + (−1 − i)6-s + 2i·7-s + (−2 + 2i)8-s + 9-s − 2i·12-s − 4·13-s + (−2 + 2i)14-s − 4·16-s + 2i·17-s + (1 + i)18-s + 4i·19-s − 2i·21-s + ⋯ |
| L(s) = 1 | + (0.707 + 0.707i)2-s − 0.577·3-s + i·4-s + (−0.408 − 0.408i)6-s + 0.755i·7-s + (−0.707 + 0.707i)8-s + 0.333·9-s − 0.577i·12-s − 1.10·13-s + (−0.534 + 0.534i)14-s − 16-s + 0.485i·17-s + (0.235 + 0.235i)18-s + 0.917i·19-s − 0.436i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.208277 + 1.28346i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.208277 + 1.28346i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−11.37331684323367922348054912130, −10.16937877639622175159315053319, −9.236559307019962860025346071624, −8.154389238982362704566127887242, −7.39455817417441250097264741188, −6.33381802077517724086867331514, −5.61505272819803059433681440237, −4.80713801216718580854087167474, −3.66080884828571460979894058315, −2.26658366836664787218179475693,
0.59793390921578088501730731337, 2.27687552175324204583534412412, 3.59109641256330707808232607961, 4.74817072293679982607834694681, 5.27213745154939131407998913986, 6.69363539304346412974409166717, 7.16239952312079613426910753772, 8.743380302330356769452213775578, 9.829932887489820091776894785556, 10.42438477100925057134788537796
