L(s) = 1 | + (−1.42 + 1.72i)5-s + (−3.11 + 3.11i)7-s − 1.17·11-s + (2.15 − 2.15i)13-s + (−1.33 − 1.33i)17-s − 0.322·19-s + (−4.71 + 4.71i)23-s + (−0.933 − 4.91i)25-s − 6.63i·29-s + 0.0675·31-s + (−0.924 − 9.82i)35-s + (7.60 + 7.60i)37-s + 3.19i·41-s + (6.70 − 6.70i)43-s + (−7.34 − 7.34i)47-s + ⋯ |
L(s) = 1 | + (−0.637 + 0.770i)5-s + (−1.17 + 1.17i)7-s − 0.355·11-s + (0.596 − 0.596i)13-s + (−0.323 − 0.323i)17-s − 0.0739·19-s + (−0.982 + 0.982i)23-s + (−0.186 − 0.982i)25-s − 1.23i·29-s + 0.0121·31-s + (−0.156 − 1.66i)35-s + (1.25 + 1.25i)37-s + 0.499i·41-s + (1.02 − 1.02i)43-s + (−1.07 − 1.07i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1914503257\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1914503257\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.42 - 1.72i)T \) |
good | 7 | \( 1 + (3.11 - 3.11i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.17T + 11T^{2} \) |
| 13 | \( 1 + (-2.15 + 2.15i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.33 + 1.33i)T + 17iT^{2} \) |
| 19 | \( 1 + 0.322T + 19T^{2} \) |
| 23 | \( 1 + (4.71 - 4.71i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.63iT - 29T^{2} \) |
| 31 | \( 1 - 0.0675T + 31T^{2} \) |
| 37 | \( 1 + (-7.60 - 7.60i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.19iT - 41T^{2} \) |
| 43 | \( 1 + (-6.70 + 6.70i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.34 + 7.34i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.73 + 5.73i)T + 53iT^{2} \) |
| 59 | \( 1 + 8.68iT - 59T^{2} \) |
| 61 | \( 1 + 12.5iT - 61T^{2} \) |
| 67 | \( 1 + (1.87 + 1.87i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.18iT - 71T^{2} \) |
| 73 | \( 1 + (-3.97 - 3.97i)T + 73iT^{2} \) |
| 79 | \( 1 + 9.66iT - 79T^{2} \) |
| 83 | \( 1 + (0.585 + 0.585i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.557T + 89T^{2} \) |
| 97 | \( 1 + (10.5 - 10.5i)T - 97iT^{2} \) |
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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
−9.492520006831729346942734459608, −8.287364074241284296681739817914, −7.85217041420873014336908099842, −6.60954763173598336750463562978, −6.17914086082742343792507638132, −5.24930120209212671014361770279, −3.88189939512778751755805584604, −3.12054152364765058189276469279, −2.30068182743373958518914733924, −0.082657708418151495063797115503,
1.20653498881628110624001346558, 2.89857004783762032213231809326, 4.09528453280786800058028584155, 4.31335500101890702074596555394, 5.77084193159628772159897102415, 6.57596775034936231053462438370, 7.39201317010844985774304760979, 8.141847739577482907660625227479, 9.050106507646774605214393121881, 9.662045278649707577344423334282