| L(s) = 1 | − i·2-s + 3-s − 4-s − i·6-s + 4.60i·7-s + i·8-s + 9-s − 12-s − 3.60·13-s + 4.60·14-s + 16-s − 4.60·17-s − i·18-s − 4.60i·19-s + 4.60i·21-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.408i·6-s + 1.74i·7-s + 0.353i·8-s + 0.333·9-s − 0.288·12-s − 1.00·13-s + 1.23·14-s + 0.250·16-s − 1.11·17-s − 0.235i·18-s − 1.05i·19-s + 1.00i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.102690392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.102690392\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 3.60T \) |
| good | 7 | \( 1 - 4.60iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 + 4.60iT - 19T^{2} \) |
| 23 | \( 1 - 1.39T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 - 9.21iT - 37T^{2} \) |
| 41 | \( 1 - 3.21iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 9.21iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 9.21iT - 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 3.21iT - 67T^{2} \) |
| 71 | \( 1 - 9.21iT - 71T^{2} \) |
| 73 | \( 1 - 1.39iT - 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 2.78iT - 83T^{2} \) |
| 89 | \( 1 - 15.2iT - 89T^{2} \) |
| 97 | \( 1 + 1.39iT - 97T^{2} \) |
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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
−9.400471241018975842579187616968, −8.656618550307266208788710844513, −8.269176950484732283536046969473, −7.01296318327460675591349571637, −6.22420389064173470069771826794, −4.95922689061128231617247030263, −4.67349646083220932559689189366, −3.00251616478735544647298974404, −2.70613641588438421311108869219, −1.67856806084251347399750267055,
0.34965443300621170534743459332, 1.87600191227419803787448943768, 3.29193333186213728188301094203, 4.20843723232626618347097079893, 4.69797069752167179308593187933, 5.95346426375846843507057266931, 6.94592582506345169153678465371, 7.34710306672406512673244208951, 8.031395346640823324503469863836, 8.865163127212685297150616758217
