L(s) = 1 | + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)13-s + (0.866 − 0.499i)15-s + (−0.5 + 0.866i)17-s − i·19-s + (−0.866 + 0.5i)23-s + i·27-s + (0.5 + 0.866i)29-s + 2i·31-s − 0.999i·39-s + (−0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.866 − 0.5i)47-s + 49-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)13-s + (0.866 − 0.499i)15-s + (−0.5 + 0.866i)17-s − i·19-s + (−0.866 + 0.5i)23-s + i·27-s + (0.5 + 0.866i)29-s + 2i·31-s − 0.999i·39-s + (−0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.866 − 0.5i)47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5788813692\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5788813692\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + iT \) |
good | 3 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - 2iT - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - 2iT - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.45468339489145265525669036034, −9.168610146115285180020975984204, −8.496181170385586814556598231387, −7.29342028013455314973685785391, −6.75182038799543721294901214089, −6.18390062174837916117022491537, −5.10737313499839080746870562739, −3.98271308439624285514057492539, −3.01058750954397814600227403540, −1.50863776185989750114429940837,
0.59137151464952032825805018407, 2.46690302912986844679491042576, 4.05246587633755526302103191773, 4.51414379946950718010219316691, 5.66622558563739064601229591529, 6.00095206692150054700176195975, 7.46601906877010043582567352113, 8.165910218485287290731966057280, 8.905033504640740222363836905380, 9.968958838414498716776928380114