| L(s) = 1 | + i·2-s − 4-s + 2.23i·5-s − 3.23i·7-s − i·8-s − 2.23·10-s − 2·11-s + 0.763i·13-s + 3.23·14-s + 16-s + i·17-s − 2.47·19-s − 2.23i·20-s − 2i·22-s − 4i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.999i·5-s − 1.22i·7-s − 0.353i·8-s − 0.707·10-s − 0.603·11-s + 0.211i·13-s + 0.864·14-s + 0.250·16-s + 0.242i·17-s − 0.567·19-s − 0.499i·20-s − 0.426i·22-s − 0.834i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3992252965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3992252965\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.23iT \) |
| 17 | \( 1 - iT \) |
| good | 7 | \( 1 + 3.23iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 0.763iT - 13T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 5.70T + 41T^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 - 1.52iT - 47T^{2} \) |
| 53 | \( 1 + 6.94iT - 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 11.7iT - 67T^{2} \) |
| 71 | \( 1 + 6.76T + 71T^{2} \) |
| 73 | \( 1 - 13.2iT - 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 1.52iT - 83T^{2} \) |
| 89 | \( 1 - 3.52T + 89T^{2} \) |
| 97 | \( 1 + 11.7iT - 97T^{2} \) |
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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
−9.210854202503868203551152554487, −8.233093154233786589490847563469, −7.45256697054074903015477362962, −6.95612248995389147347192175370, −6.20874856829200788660934405213, −5.20865955361717963550896491072, −4.12332053070412539081218670458, −3.44729324900800156391099060520, −2.05322668054936673380114053844, −0.15128576267795183587833397861,
1.53227414844028222275862722486, 2.49479410890826440591981969838, 3.57879529500617047421428488749, 4.73657189795930188027600953159, 5.38553090601852159792908910602, 6.07082410406131634535431622837, 7.55452545554919402999044052426, 8.274554493553103265466153121494, 9.073191396631734250917509722633, 9.444485548926707973619445412577
