L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + (0.5 − 0.866i)9-s − 2·13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.5 − 0.866i)25-s + (−1 + 1.73i)26-s + (0.499 + 0.866i)32-s − 0.999·34-s − 0.999·36-s − 0.999·50-s + (0.999 + 1.73i)52-s + (−1 − 1.73i)53-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + (0.5 − 0.866i)9-s − 2·13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.5 − 0.866i)25-s + (−1 + 1.73i)26-s + (0.499 + 0.866i)32-s − 0.999·34-s − 0.999·36-s − 0.999·50-s + (0.999 + 1.73i)52-s + (−1 − 1.73i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9486639921\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9486639921\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + 2T + T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 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
−8.666240368694767713621973908942, −7.57318488192877308073674614286, −6.83823250939125812839740791210, −6.10221722846231853893072596094, −5.00527919875885398536363600171, −4.61226772355878742124274408018, −3.65039299881290295411485060490, −2.72259065586524609983632754041, −1.95152508359678581101723423990, −0.44385434784124211530261702712,
1.97053740060680400843865440243, 2.89528181419206752831906828286, 4.05679493636604253725648459282, 4.72767493214349438419493053847, 5.31762033894491866688723147196, 6.19367765564667667915382483722, 7.10413381048564031751491802937, 7.55801255485593888238791729295, 8.147287188195646889761782045648, 9.118221012211938747699426161632