| L(s) = 1 | + 3.23i·3-s − 2.23·5-s + 0.763i·7-s − 7.47·9-s − 7.23i·15-s − 2.47·21-s + 5.70i·23-s + 5.00·25-s − 14.4i·27-s − 6·29-s − 1.70i·35-s − 4.47·41-s + 11.2i·43-s + 16.7·45-s + 13.7i·47-s + ⋯ |
| L(s) = 1 | + 1.86i·3-s − 0.999·5-s + 0.288i·7-s − 2.49·9-s − 1.86i·15-s − 0.539·21-s + 1.19i·23-s + 1.00·25-s − 2.78i·27-s − 1.11·29-s − 0.288i·35-s − 0.698·41-s + 1.71i·43-s + 2.49·45-s + 1.99i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(-0.765918i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.765918i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| good | 3 | \( 1 - 3.23iT - 3T^{2} \) |
| 7 | \( 1 - 0.763iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.70iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 13.7iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 8.18iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 17.7iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.54081180536531209042938298204, −11.28380129977894010022732637833, −10.20543804354717127961279603811, −9.375059098659536273399526608005, −8.595704134806719994888197460077, −7.55268921339908238736794427242, −5.88291129271952006489432174005, −4.86724775767359282115239252340, −3.96393813181188993381959928447, −3.07259212614449354672858705353,
0.54203674593918975402133761281, 2.21309292232618562980695288057, 3.68197503671921374509558661383, 5.37405656422761824414783060441, 6.70028884169748980067068633430, 7.22137510876829682312366512447, 8.145333263117380176796508711167, 8.799248625798489187961820827836, 10.55075035709947792383124374864, 11.52973018239200456695984912183
