L(s) = 1 | + (−1.34 − 0.430i)2-s + (−0.916 + 1.46i)3-s + (1.62 + 1.15i)4-s + 0.348i·5-s + (1.86 − 1.58i)6-s + (−1.69 − 2.26i)8-s + (−1.31 − 2.69i)9-s + (0.150 − 0.469i)10-s + 3.90·11-s + (−3.19 + 1.33i)12-s + 2.93·13-s + (−0.512 − 0.319i)15-s + (1.30 + 3.77i)16-s − 3.90i·17-s + (0.617 + 4.19i)18-s − 5.57i·19-s + ⋯ |
L(s) = 1 | + (−0.952 − 0.304i)2-s + (−0.529 + 0.848i)3-s + (0.814 + 0.579i)4-s + 0.155i·5-s + (0.762 − 0.647i)6-s + (−0.599 − 0.800i)8-s + (−0.439 − 0.898i)9-s + (0.0474 − 0.148i)10-s + 1.17·11-s + (−0.923 + 0.384i)12-s + 0.815·13-s + (−0.132 − 0.0825i)15-s + (0.327 + 0.944i)16-s − 0.946i·17-s + (0.145 + 0.989i)18-s − 1.27i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.841477 + 0.168178i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.841477 + 0.168178i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.34 + 0.430i)T \) |
| 3 | \( 1 + (0.916 - 1.46i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 0.348iT - 5T^{2} \) |
| 11 | \( 1 - 3.90T + 11T^{2} \) |
| 13 | \( 1 - 2.93T + 13T^{2} \) |
| 17 | \( 1 + 3.90iT - 17T^{2} \) |
| 19 | \( 1 + 5.57iT - 19T^{2} \) |
| 23 | \( 1 + 2.18T + 23T^{2} \) |
| 29 | \( 1 - 9.75iT - 29T^{2} \) |
| 31 | \( 1 - 2.63iT - 31T^{2} \) |
| 37 | \( 1 - 0.639T + 37T^{2} \) |
| 41 | \( 1 - 7.57iT - 41T^{2} \) |
| 43 | \( 1 + 2.51iT - 43T^{2} \) |
| 47 | \( 1 - 4.36T + 47T^{2} \) |
| 53 | \( 1 + 1.72iT - 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 0.639iT - 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 7.87T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 10.5iT - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.80185870370318531780962295440, −9.843202196031929901510600073453, −9.060972124532225376115470915802, −8.593206489220664910359634975313, −6.98927473366095527089903045003, −6.54129989633316213576650785525, −5.18525656580200964063387141675, −3.91978428942513463134435027319, −2.93167491781606170369917519936, −1.01323756932588548871768424696,
1.01214655487073257958045288397, 2.08605698565400616673846815585, 3.96146835490492033259436920909, 5.70733703531021994093981692353, 6.19864574782005998444724138180, 7.05814228349927325586978121785, 8.118552705705232208991558702550, 8.600331494346146612292124015765, 9.763405943289541827430707475700, 10.61987856022080171337965278249