| L(s) = 1 | + 2-s + 4-s + i·5-s − i·7-s + 8-s + i·10-s + (2.68 + 1.95i)11-s − 3.80i·13-s − i·14-s + 16-s + 4.16·17-s + 4.38i·19-s + i·20-s + (2.68 + 1.95i)22-s + 1.92i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447i·5-s − 0.377i·7-s + 0.353·8-s + 0.316i·10-s + (0.808 + 0.588i)11-s − 1.05i·13-s − 0.267i·14-s + 0.250·16-s + 1.01·17-s + 1.00i·19-s + 0.223i·20-s + (0.571 + 0.416i)22-s + 0.402i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.691802394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.691802394\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-2.68 - 1.95i)T \) |
| good | 13 | \( 1 + 3.80iT - 13T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 19 | \( 1 - 4.38iT - 19T^{2} \) |
| 23 | \( 1 - 1.92iT - 23T^{2} \) |
| 29 | \( 1 - 3.26T + 29T^{2} \) |
| 31 | \( 1 + 2.75T + 31T^{2} \) |
| 37 | \( 1 - 5.39T + 37T^{2} \) |
| 41 | \( 1 - 1.50T + 41T^{2} \) |
| 43 | \( 1 + 4.93iT - 43T^{2} \) |
| 47 | \( 1 + 2.42iT - 47T^{2} \) |
| 53 | \( 1 - 9.74iT - 53T^{2} \) |
| 59 | \( 1 + 14.9iT - 59T^{2} \) |
| 61 | \( 1 + 2.83iT - 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 9.72iT - 71T^{2} \) |
| 73 | \( 1 - 13.5iT - 73T^{2} \) |
| 79 | \( 1 - 1.20iT - 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 + 3.44iT - 89T^{2} \) |
| 97 | \( 1 - 6.80T + 97T^{2} \) |
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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
−7.68436518224419936593190357535, −7.36681346526591043190797903226, −6.46846569536845332157481823885, −5.86808873443070998750381956327, −5.21756217194005736051701061230, −4.29117622954941587242593266549, −3.61704826762856365974747464833, −3.03101524323141035482248578902, −1.95131167408848267057357273261, −0.982389900546789410647228212056,
0.862439119015883764487462906365, 1.83173184724884566261604795956, 2.82586298071837894221503267133, 3.58508376962482883978833564590, 4.47123925602575157369049105252, 4.89797526336598623131736372606, 5.94977244633743698559053064214, 6.25873554506744400683047212716, 7.12846176516084560201350170523, 7.81106426569328031398388652228
