| L(s) = 1 | − 2-s − i·3-s + 4-s − 3.74·5-s + i·6-s + (1.87 − 1.87i)7-s − 8-s − 9-s + 3.74·10-s − i·12-s + 2i·13-s + (−1.87 + 1.87i)14-s + 3.74i·15-s + 16-s − 3.74·17-s + 18-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.5·4-s − 1.67·5-s + 0.408i·6-s + (0.707 − 0.707i)7-s − 0.353·8-s − 0.333·9-s + 1.18·10-s − 0.288i·12-s + 0.554i·13-s + (−0.499 + 0.499i)14-s + 0.966i·15-s + 0.250·16-s − 0.907·17-s + 0.235·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.109 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.241879 + 0.216729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.241879 + 0.216729i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (-1.87 + 1.87i)T \) |
| 23 | \( 1 + (3 + 3.74i)T \) |
| good | 5 | \( 1 + 3.74T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 3.74T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 3.74iT - 53T^{2} \) |
| 59 | \( 1 - 14iT - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 3.74iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 3.74iT - 79T^{2} \) |
| 83 | \( 1 - 7.48T + 83T^{2} \) |
| 89 | \( 1 + 3.74T + 89T^{2} \) |
| 97 | \( 1 + 7.48T + 97T^{2} \) |
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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
−10.50780825716280140566530394373, −9.087181902153083853352377664978, −8.243465216488187656794552438259, −7.939017020320324142239970168965, −6.99407980810050744587776477388, −6.44128523043874786080472940849, −4.65255169434765675738897007516, −4.06970718817477199853871097078, −2.64785504253161550235915170696, −1.16519377744410006594611255940,
0.21565097265982175410807593931, 2.22944724104257325132027623310, 3.52349741632011618528708475073, 4.39299693005703384507156087403, 5.37300817812725155917876589055, 6.61099756914441127167519122422, 7.62456831478798970524028980756, 8.371779610317601768403738880905, 8.669596423638359344771920696123, 9.804869452343042670478502149506
