L(s) = 1 | + (−2.10 + 2.10i)5-s − 4.40·7-s + (0.215 + 0.215i)11-s + (2.73 − 2.73i)13-s − 2.36i·17-s + (−0.758 − 0.758i)19-s + 1.75i·23-s − 3.86i·25-s + (5.54 + 5.54i)29-s − 9.01i·31-s + (9.27 − 9.27i)35-s + (−3.10 − 3.10i)37-s + 10.1·41-s + (3.54 − 3.54i)43-s + 3.90·47-s + ⋯ |
L(s) = 1 | + (−0.941 + 0.941i)5-s − 1.66·7-s + (0.0650 + 0.0650i)11-s + (0.758 − 0.758i)13-s − 0.573i·17-s + (−0.174 − 0.174i)19-s + 0.366i·23-s − 0.772i·25-s + (1.02 + 1.02i)29-s − 1.61i·31-s + (1.56 − 1.56i)35-s + (−0.510 − 0.510i)37-s + 1.57·41-s + (0.540 − 0.540i)43-s + 0.569·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7911569046\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7911569046\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.10 - 2.10i)T - 5iT^{2} \) |
| 7 | \( 1 + 4.40T + 7T^{2} \) |
| 11 | \( 1 + (-0.215 - 0.215i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.73 + 2.73i)T - 13iT^{2} \) |
| 17 | \( 1 + 2.36iT - 17T^{2} \) |
| 19 | \( 1 + (0.758 + 0.758i)T + 19iT^{2} \) |
| 23 | \( 1 - 1.75iT - 23T^{2} \) |
| 29 | \( 1 + (-5.54 - 5.54i)T + 29iT^{2} \) |
| 31 | \( 1 + 9.01iT - 31T^{2} \) |
| 37 | \( 1 + (3.10 + 3.10i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + (-3.54 + 3.54i)T - 43iT^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 + (-2.71 + 2.71i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.40 + 3.40i)T + 59iT^{2} \) |
| 61 | \( 1 + (-1.75 + 1.75i)T - 61iT^{2} \) |
| 67 | \( 1 + (9.11 + 9.11i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 - 0.482iT - 73T^{2} \) |
| 79 | \( 1 - 6.88iT - 79T^{2} \) |
| 83 | \( 1 + (4.79 - 4.79i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.00T + 89T^{2} \) |
| 97 | \( 1 + 3.34T + 97T^{2} \) |
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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
−9.683233825834395170528487847190, −8.972471008821949923749124915349, −7.85284762719194779579822779522, −7.15417467104354003007615789143, −6.44475986082914161511367928974, −5.65045941457947302062463721998, −4.12623879689587553022592120048, −3.36464225285713083830308551842, −2.70945029373442735440338302335, −0.43187648587952044593815539153,
1.02376243637053705633121838487, 2.84332137193112991560551601261, 3.90038451475089960666507939600, 4.45599518991280293687305989849, 5.87185739477175271174945519279, 6.51354116930437905367145022526, 7.43261088506525176657460595915, 8.585487457402118051373663915377, 8.857128416532135995166687410190, 9.876530307747851128680709477988