| L(s) = 1 | − 2.69·2-s + 5.24·4-s − 1.40·5-s − 8.73·8-s + 3.78·10-s − 6.04·11-s − 13-s + 13.0·16-s − 6.70·17-s − 1.53·19-s − 7.36·20-s + 16.2·22-s − 0.742·23-s − 3.02·25-s + 2.69·26-s + 6.12·29-s − 10.0·31-s − 17.5·32-s + 18.0·34-s − 6.49·37-s + 4.13·38-s + 12.2·40-s − 7.53·41-s − 1.53·43-s − 31.7·44-s + 2·46-s + 5.30·47-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 2.62·4-s − 0.628·5-s − 3.08·8-s + 1.19·10-s − 1.82·11-s − 0.277·13-s + 3.25·16-s − 1.62·17-s − 0.352·19-s − 1.64·20-s + 3.46·22-s − 0.154·23-s − 0.605·25-s + 0.527·26-s + 1.13·29-s − 1.80·31-s − 3.10·32-s + 3.09·34-s − 1.06·37-s + 0.670·38-s + 1.94·40-s − 1.17·41-s − 0.234·43-s − 4.78·44-s + 0.294·46-s + 0.773·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01998638417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01998638417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 5 | \( 1 + 1.40T + 5T^{2} \) |
| 11 | \( 1 + 6.04T + 11T^{2} \) |
| 17 | \( 1 + 6.70T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 + 0.742T + 23T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 6.49T + 37T^{2} \) |
| 41 | \( 1 + 7.53T + 41T^{2} \) |
| 43 | \( 1 + 1.53T + 43T^{2} \) |
| 47 | \( 1 - 5.30T + 47T^{2} \) |
| 53 | \( 1 - 5.96T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 4.49T + 67T^{2} \) |
| 71 | \( 1 - 3.23T + 71T^{2} \) |
| 73 | \( 1 + 8.02T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 8.11T + 83T^{2} \) |
| 89 | \( 1 - 1.24T + 89T^{2} \) |
| 97 | \( 1 - 0.464T + 97T^{2} \) |
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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
−8.330143847664147743726458413895, −7.42871964359471971851873902926, −7.23984091133441246464982910378, −6.31665985656797558334725301992, −5.46943586095419010942587917346, −4.47949873378708745483359821481, −3.23669862050327215857336425962, −2.43631639811079153421945237785, −1.74560179043372865993383051424, −0.093529032624005715487122135163,
0.093529032624005715487122135163, 1.74560179043372865993383051424, 2.43631639811079153421945237785, 3.23669862050327215857336425962, 4.47949873378708745483359821481, 5.46943586095419010942587917346, 6.31665985656797558334725301992, 7.23984091133441246464982910378, 7.42871964359471971851873902926, 8.330143847664147743726458413895
