L(s) = 1 | + (0.365 + 0.930i)2-s + (0.0111 + 0.149i)3-s + (−0.733 + 0.680i)4-s + (−0.134 + 0.0648i)6-s + (−0.900 − 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.966 − 0.145i)9-s + (−0.722 − 0.108i)11-s + (−0.109 − 0.101i)12-s + (0.455 − 0.571i)13-s + (0.0747 − 0.997i)14-s + (0.0747 − 0.997i)16-s + (0.955 − 0.294i)17-s + (0.488 + 0.846i)18-s + (0.0546 − 0.139i)21-s + (−0.162 − 0.712i)22-s + ⋯ |
L(s) = 1 | + (0.365 + 0.930i)2-s + (0.0111 + 0.149i)3-s + (−0.733 + 0.680i)4-s + (−0.134 + 0.0648i)6-s + (−0.900 − 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.966 − 0.145i)9-s + (−0.722 − 0.108i)11-s + (−0.109 − 0.101i)12-s + (0.455 − 0.571i)13-s + (0.0747 − 0.997i)14-s + (0.0747 − 0.997i)16-s + (0.955 − 0.294i)17-s + (0.488 + 0.846i)18-s + (0.0546 − 0.139i)21-s + (−0.162 − 0.712i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.357073659\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.357073659\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.365 - 0.930i)T \) |
| 7 | \( 1 + (0.900 + 0.433i)T \) |
| 17 | \( 1 + (-0.955 + 0.294i)T \) |
good | 3 | \( 1 + (-0.0111 - 0.149i)T + (-0.988 + 0.149i)T^{2} \) |
| 5 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 11 | \( 1 + (0.722 + 0.108i)T + (0.955 + 0.294i)T^{2} \) |
| 13 | \( 1 + (-0.455 + 0.571i)T + (-0.222 - 0.974i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.19 - 0.367i)T + (0.826 + 0.563i)T^{2} \) |
| 29 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 41 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 53 | \( 1 + (1.21 - 1.12i)T + (0.0747 - 0.997i)T^{2} \) |
| 59 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 61 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.367 + 1.61i)T + (-0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 79 | \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.147 - 0.0222i)T + (0.955 - 0.294i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.871770660845199918585845369735, −7.941059994532617736886443477543, −7.40086708170744786126659188851, −6.67380836451450096010431793120, −6.05704787059300114566197387993, −5.08881642213373726822315305258, −4.51917911135823949756279226411, −3.38005738007319336524906856408, −3.03050863900390252097999584342, −0.956515605941062994820044242250,
1.08062908940377190063004844794, 2.20176569876881827714066831947, 3.07186736988142711478261935602, 3.84855105094331348779638038421, 4.74576244018889608123623572431, 5.51915073472987302290614914357, 6.32228618599948798454784381683, 7.09034467106052180312589423108, 8.069390745390431860098621760174, 8.877032040897040853883108193363