L(s) = 1 | + (−0.707 + 0.707i)3-s + (2.23 + 0.162i)5-s + 2.93·7-s − 1.00i·9-s + (−0.663 + 0.663i)11-s + (1.12 − 1.12i)13-s + (−1.69 + 1.46i)15-s − 7.47i·17-s + (0.423 + 0.423i)19-s + (−2.07 + 2.07i)21-s + 6.17·23-s + (4.94 + 0.722i)25-s + (0.707 + 0.707i)27-s + (2.95 + 2.95i)29-s − 1.82·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.997 + 0.0724i)5-s + 1.10·7-s − 0.333i·9-s + (−0.200 + 0.200i)11-s + (0.312 − 0.312i)13-s + (−0.436 + 0.377i)15-s − 1.81i·17-s + (0.0971 + 0.0971i)19-s + (−0.453 + 0.453i)21-s + 1.28·23-s + (0.989 + 0.144i)25-s + (0.136 + 0.136i)27-s + (0.548 + 0.548i)29-s − 0.327·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.161834121\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.161834121\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-2.23 - 0.162i)T \) |
good | 7 | \( 1 - 2.93T + 7T^{2} \) |
| 11 | \( 1 + (0.663 - 0.663i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.12 + 1.12i)T - 13iT^{2} \) |
| 17 | \( 1 + 7.47iT - 17T^{2} \) |
| 19 | \( 1 + (-0.423 - 0.423i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.17T + 23T^{2} \) |
| 29 | \( 1 + (-2.95 - 2.95i)T + 29iT^{2} \) |
| 31 | \( 1 + 1.82T + 31T^{2} \) |
| 37 | \( 1 + (5.53 + 5.53i)T + 37iT^{2} \) |
| 41 | \( 1 + 12.3iT - 41T^{2} \) |
| 43 | \( 1 + (0.897 + 0.897i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.12iT - 47T^{2} \) |
| 53 | \( 1 + (-0.146 - 0.146i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.72 - 7.72i)T - 59iT^{2} \) |
| 61 | \( 1 + (-7.37 - 7.37i)T + 61iT^{2} \) |
| 67 | \( 1 + (8.68 - 8.68i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.95iT - 71T^{2} \) |
| 73 | \( 1 + 0.174T + 73T^{2} \) |
| 79 | \( 1 + 3.06T + 79T^{2} \) |
| 83 | \( 1 + (-9.18 + 9.18i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.71iT - 89T^{2} \) |
| 97 | \( 1 + 10.5iT - 97T^{2} \) |
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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
−9.008035993340987144639341540468, −8.807714517774925652891609042736, −7.38314206206976356051381478618, −6.97032708643673391091534797176, −5.66555364994878783641593057330, −5.23961202242926819267390252804, −4.57366826335221965583794139277, −3.21691158356314889875480355323, −2.18968646170737906512590395325, −0.969627276880556157495418460119,
1.28837671646428081432978838786, 1.89576259970697403956078191923, 3.19816026989014588092920336146, 4.61904951263272088301553495406, 5.14584787738815090084207716767, 6.18846747273248286934639603319, 6.54432968689775583841725672253, 7.79865344353755844860922923465, 8.345476938080849233275026149644, 9.129992779748695070040771495456