L(s) = 1 | − 3.87·5-s + (2 − 1.73i)7-s + 2.23i·11-s − 3.46i·13-s + 5.19i·19-s + 2.23i·23-s + 10.0·25-s + 4.47i·29-s − 1.73i·31-s + (−7.74 + 6.70i)35-s − 37-s − 3.87·41-s − 2·43-s + 7.74·47-s + (1.00 − 6.92i)49-s + ⋯ |
L(s) = 1 | − 1.73·5-s + (0.755 − 0.654i)7-s + 0.674i·11-s − 0.960i·13-s + 1.19i·19-s + 0.466i·23-s + 2.00·25-s + 0.830i·29-s − 0.311i·31-s + (−1.30 + 1.13i)35-s − 0.164·37-s − 0.604·41-s − 0.304·43-s + 1.12·47-s + (0.142 − 0.989i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.150925197\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.150925197\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 5 | \( 1 + 3.87T + 5T^{2} \) |
| 11 | \( 1 - 2.23iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 2.23iT - 23T^{2} \) |
| 29 | \( 1 - 4.47iT - 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 3.87T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 7.74T + 47T^{2} \) |
| 53 | \( 1 + 8.94iT - 53T^{2} \) |
| 59 | \( 1 + 7.74T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 11.1iT - 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 7.74T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−8.261075618462827604632706075102, −7.85420956556942570740475803647, −7.42203594873131298531163985876, −6.57167038283071507480727933361, −5.29536850351405197216663924773, −4.67943542824686087396002996831, −3.79169159630839195578426690169, −3.32825425642453255345432512781, −1.77288932349726725270360491163, −0.52151947117597763379591725628,
0.832079874955314230563478105373, 2.32824030452573480829642343006, 3.27906189360233406784391475590, 4.27135923135635035578956722115, 4.69124474677935603357246577137, 5.72905741876912844479151047101, 6.76205607996479939640679517000, 7.38754986501960877967527165172, 8.176166886012653517981589613883, 8.683430344080710952627967999922