L(s) = 1 | + 3-s + 1.72i·5-s + (2.63 − 0.222i)7-s + 9-s + 6.17i·11-s − 2.82i·13-s + 1.72i·15-s + 1.72i·17-s − 5.90·19-s + (2.63 − 0.222i)21-s − 1.54i·23-s + 2.01·25-s + 27-s + 8.28·29-s − 4.62·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.772i·5-s + (0.996 − 0.0839i)7-s + 0.333·9-s + 1.86i·11-s − 0.784i·13-s + 0.446i·15-s + 0.419i·17-s − 1.35·19-s + (0.575 − 0.0484i)21-s − 0.322i·23-s + 0.402·25-s + 0.192·27-s + 1.53·29-s − 0.831·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.78747 + 0.830046i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78747 + 0.830046i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + (-2.63 + 0.222i)T \) |
good | 5 | \( 1 - 1.72iT - 5T^{2} \) |
| 11 | \( 1 - 6.17iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 1.72iT - 17T^{2} \) |
| 19 | \( 1 + 5.90T + 19T^{2} \) |
| 23 | \( 1 + 1.54iT - 23T^{2} \) |
| 29 | \( 1 - 8.28T + 29T^{2} \) |
| 31 | \( 1 + 4.62T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 + 3.92iT - 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 - 4.88T + 47T^{2} \) |
| 53 | \( 1 + 9.17T + 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 + 14.2iT - 61T^{2} \) |
| 67 | \( 1 - 6.64iT - 67T^{2} \) |
| 71 | \( 1 + 5.00iT - 71T^{2} \) |
| 73 | \( 1 - 2.19iT - 73T^{2} \) |
| 79 | \( 1 + 9.55iT - 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 9.16iT - 89T^{2} \) |
| 97 | \( 1 + 17.1iT - 97T^{2} \) |
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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
−10.48639902129966050455285515679, −9.971460606208847162003545321272, −8.769796565661272033539943329129, −7.971397217151675363249551590115, −7.23274271973602203545682488095, −6.37967152947484092040596002930, −4.90306474515578374695848188095, −4.18785379853488530109142632889, −2.75015643463081279282559512106, −1.79950512818282886491283119469,
1.10515141901177421920431499057, 2.50090587673871485689066299179, 3.87711647648630139783688324353, 4.79427270100469632042395873401, 5.78807747692425263044720543962, 6.91157329428232996672360992649, 8.170017348563640631442494056281, 8.598678471663449436225664965461, 9.152437517870262720892920255849, 10.49720200608559343085033718750