| L(s) = 1 | − i·2-s − 4-s − i·7-s + i·8-s − 2·11-s − i·13-s − 14-s + 16-s + i·17-s − 4·19-s + 2i·22-s + 7i·23-s − 26-s + i·28-s + 29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.377i·7-s + 0.353i·8-s − 0.603·11-s − 0.277i·13-s − 0.267·14-s + 0.250·16-s + 0.242i·17-s − 0.917·19-s + 0.426i·22-s + 1.45i·23-s − 0.196·26-s + 0.188i·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.098415133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.098415133\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 7iT - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 11iT - 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 3iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 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
−8.902186558754098285799163241864, −7.917622718233711942763675610896, −7.52875966659886035518224209237, −6.36490624378062815697104960504, −5.62609604427547737567948586981, −4.71876505107304421530976423227, −3.96838017352575114936592203683, −3.08103096442649605749197867140, −2.17203488202553299466463835934, −1.03404631717384617889362410988,
0.38891980352997469944515553764, 2.04139695684873212623534625709, 2.98695522803573316007910573951, 4.16767890302369733025766307495, 4.85295102670311579135486305208, 5.62442206046540222346720984564, 6.51388161801017148728456699899, 6.93824515043766526694219211083, 8.056352970138622546857262447687, 8.441209391517775762363458788091
