L(s) = 1 | − i·5-s + 2.44i·7-s − 5.91·11-s − 6.24·13-s + 4.89i·19-s + 1.43·23-s − 25-s − 6i·29-s + 1.43i·31-s + 2.44·35-s − 2.24·37-s − 4.24i·41-s + 11.8i·43-s − 11.8·47-s + 1.00·49-s + ⋯ |
L(s) = 1 | − 0.447i·5-s + 0.925i·7-s − 1.78·11-s − 1.73·13-s + 1.12i·19-s + 0.299·23-s − 0.200·25-s − 1.11i·29-s + 0.257i·31-s + 0.414·35-s − 0.368·37-s − 0.662i·41-s + 1.80i·43-s − 1.72·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0704155 + 0.321128i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0704155 + 0.321128i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
good | 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 11 | \( 1 + 5.91T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 1.43iT - 31T^{2} \) |
| 37 | \( 1 + 2.24T + 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 11.8iT - 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 8.48iT - 53T^{2} \) |
| 59 | \( 1 + 5.91T + 59T^{2} \) |
| 61 | \( 1 + 6.48T + 61T^{2} \) |
| 67 | \( 1 + 6.92iT - 67T^{2} \) |
| 71 | \( 1 + 2.86T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 + 8.36iT - 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 7.75iT - 89T^{2} \) |
| 97 | \( 1 - 6.48T + 97T^{2} \) |
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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
−10.62285172710676401109684076759, −9.896491986292953913474383346516, −9.155714472960532050212012777137, −7.974852021687071211497299481101, −7.67243773545973836149786556195, −6.19214581425551932514295477558, −5.27405182648113633356294668182, −4.69468908140049091971446308148, −2.99808136677342464802221668659, −2.10893775451151670682347666108,
0.15369234099645180820089721213, 2.31432964550426279349159215220, 3.24264939512269243365973256496, 4.73291244164444739339383331718, 5.27509039572171596439847921976, 6.83319437780743722768346340171, 7.33448913917226378103386499267, 8.102362513437578814178625345040, 9.349061483749632971850821900289, 10.27193462296181670786664478381