L(s) = 1 | + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (0.587 + 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.951 + 0.309i)9-s + (−0.309 + 0.951i)11-s + (−0.309 − 0.951i)14-s + (0.309 + 0.951i)16-s + 0.999·18-s + (0.587 − 0.809i)22-s + 1.17i·23-s + (−0.587 + 0.809i)25-s + 0.999i·28-s + (−0.896 − 0.142i)29-s − i·32-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (0.587 + 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.951 + 0.309i)9-s + (−0.309 + 0.951i)11-s + (−0.309 − 0.951i)14-s + (0.309 + 0.951i)16-s + 0.999·18-s + (0.587 − 0.809i)22-s + 1.17i·23-s + (−0.587 + 0.809i)25-s + 0.999i·28-s + (−0.896 − 0.142i)29-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5814036764\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5814036764\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 + 0.309i)T \) |
| 7 | \( 1 + (-0.587 - 0.809i)T \) |
| 11 | \( 1 + (0.309 - 0.951i)T \) |
good | 3 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 5 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 13 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 23 | \( 1 - 1.17iT - T^{2} \) |
| 29 | \( 1 + (0.896 + 0.142i)T + (0.951 + 0.309i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.0489 - 0.309i)T + (-0.951 - 0.309i)T^{2} \) |
| 41 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + (-1.26 + 1.26i)T - iT^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.412i)T + (0.587 + 0.809i)T^{2} \) |
| 59 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 61 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 67 | \( 1 + (-1.26 - 1.26i)T + iT^{2} \) |
| 71 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T^{2} \) |
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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
−9.868569095212104834289164149008, −9.309650003108723163975319088964, −8.508645962636680145793518992159, −7.79714136862034172965942811539, −7.13579641467930547990285352228, −5.85893003870999992379743658059, −5.20844025922514531396403688759, −3.75163132483498202695067836629, −2.55524944220821024273452297855, −1.77957533918034790907030720067,
0.66850283100931345440987491752, 2.24745645311708196037540316939, 3.40353615052694851848559909397, 4.76261760735137246316541140716, 5.86061777596919473561183116497, 6.42685144369398899058085807747, 7.57674313101941731690385349760, 8.131302114088354329340032478306, 8.812587003658196704747059400391, 9.635275888500908827416973480696