L(s) = 1 | + (−0.900 − 0.433i)4-s + (0.440 + 0.0663i)5-s + (−0.955 + 1.65i)7-s + (0.0747 − 0.997i)9-s + (1.78 − 0.858i)11-s + (0.623 + 0.781i)16-s + (1.44 − 0.218i)17-s + (0.0747 + 0.997i)19-s + (−0.367 − 0.250i)20-s + (0.603 + 0.411i)23-s + (−0.766 − 0.236i)25-s + (1.57 − 1.07i)28-s + (−0.530 + 0.664i)35-s + (−0.5 + 0.866i)36-s + 43-s − 1.97·44-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)4-s + (0.440 + 0.0663i)5-s + (−0.955 + 1.65i)7-s + (0.0747 − 0.997i)9-s + (1.78 − 0.858i)11-s + (0.623 + 0.781i)16-s + (1.44 − 0.218i)17-s + (0.0747 + 0.997i)19-s + (−0.367 − 0.250i)20-s + (0.603 + 0.411i)23-s + (−0.766 − 0.236i)25-s + (1.57 − 1.07i)28-s + (−0.530 + 0.664i)35-s + (−0.5 + 0.866i)36-s + 43-s − 1.97·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8837348211\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8837348211\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 + (-0.0747 - 0.997i)T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 3 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 5 | \( 1 + (-0.440 - 0.0663i)T + (0.955 + 0.294i)T^{2} \) |
| 7 | \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.78 + 0.858i)T + (0.623 - 0.781i)T^{2} \) |
| 13 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 17 | \( 1 + (-1.44 + 0.218i)T + (0.955 - 0.294i)T^{2} \) |
| 23 | \( 1 + (-0.603 - 0.411i)T + (0.365 + 0.930i)T^{2} \) |
| 29 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 31 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 53 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 59 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 61 | \( 1 + (1.40 + 0.432i)T + (0.826 + 0.563i)T^{2} \) |
| 67 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 71 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 73 | \( 1 + (-0.455 - 1.16i)T + (-0.733 + 0.680i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (1.40 - 1.29i)T + (0.0747 - 0.997i)T^{2} \) |
| 89 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 97 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
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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
−9.953791380929106853909256343067, −9.502649516970693239647285823455, −9.070493379254891476782595987486, −8.261133580886479870094586380197, −6.59374438393545904395983698299, −5.91979985541720148029431857025, −5.53580823257021750439081329705, −3.85514639278389425209012398528, −3.19315632925824442131906684774, −1.35206971271921646929979396058,
1.27098646631467067223071798463, 3.22859805939290852114761419683, 4.12731297901676339112287174979, 4.79252097457480693883373484839, 6.16538574927837147034551191680, 7.19921062111920541742779510504, 7.66356479712478644635645816019, 8.973350889920821618843560702578, 9.695559338484607027435684945816, 10.11592710066312861490265933013