L(s) = 1 | + (0.707 + 0.707i)3-s + (−1.65 + 1.50i)5-s + 2.58·7-s + 1.00i·9-s + (−4.39 − 4.39i)11-s + (−0.417 − 0.417i)13-s + (−2.23 − 0.110i)15-s − 4.40i·17-s + (4.53 − 4.53i)19-s + (1.83 + 1.83i)21-s + 0.281·23-s + (0.494 − 4.97i)25-s + (−0.707 + 0.707i)27-s + (−3.73 + 3.73i)29-s − 3.05·31-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.741 + 0.671i)5-s + 0.978·7-s + 0.333i·9-s + (−1.32 − 1.32i)11-s + (−0.115 − 0.115i)13-s + (−0.576 − 0.0286i)15-s − 1.06i·17-s + (1.04 − 1.04i)19-s + (0.399 + 0.399i)21-s + 0.0586·23-s + (0.0989 − 0.995i)25-s + (−0.136 + 0.136i)27-s + (−0.693 + 0.693i)29-s − 0.548·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.456118416\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.456118416\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.65 - 1.50i)T \) |
good | 7 | \( 1 - 2.58T + 7T^{2} \) |
| 11 | \( 1 + (4.39 + 4.39i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.417 + 0.417i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.40iT - 17T^{2} \) |
| 19 | \( 1 + (-4.53 + 4.53i)T - 19iT^{2} \) |
| 23 | \( 1 - 0.281T + 23T^{2} \) |
| 29 | \( 1 + (3.73 - 3.73i)T - 29iT^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 + (-5.26 + 5.26i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.16iT - 41T^{2} \) |
| 43 | \( 1 + (-2.66 + 2.66i)T - 43iT^{2} \) |
| 47 | \( 1 + 7.45iT - 47T^{2} \) |
| 53 | \( 1 + (-2.89 + 2.89i)T - 53iT^{2} \) |
| 59 | \( 1 + (-4.60 - 4.60i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.211 + 0.211i)T - 61iT^{2} \) |
| 67 | \( 1 + (-7.17 - 7.17i)T + 67iT^{2} \) |
| 71 | \( 1 - 15.9iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 4.53T + 79T^{2} \) |
| 83 | \( 1 + (-4.56 - 4.56i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 6.78iT - 97T^{2} \) |
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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
−8.885848269777715486018321693058, −8.345239587817701176645855771700, −7.46269413727755420343500202962, −7.16579928963832855574749851102, −5.53975738415785289479128929139, −5.15697423425726915822257028685, −4.02318441067780211900282328928, −3.09004074337731745431319982262, −2.45202738088518338447182828208, −0.52519219505893011280195056138,
1.32761298690142267267493862632, 2.20088217708218641363205528698, 3.50982620607821572974262620291, 4.50795840886647172880312929265, 5.07978820790720602770573533900, 6.08754164084003204888353727585, 7.43801016805520622453798266014, 7.84359421628788380220473960053, 8.144143646628830016969666425584, 9.272014099614912679435709310417