| L(s) = 1 | − 3-s − 1.73i·5-s + 9-s − 1.73i·11-s + 1.73i·15-s + 3.46i·17-s + 2·19-s + 2.00·25-s − 27-s + 9·29-s − 5·31-s + 1.73i·33-s + 10·37-s − 10.3i·41-s − 3.46i·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.774i·5-s + 0.333·9-s − 0.522i·11-s + 0.447i·15-s + 0.840i·17-s + 0.458·19-s + 0.400·25-s − 0.192·27-s + 1.67·29-s − 0.898·31-s + 0.301i·33-s + 1.64·37-s − 1.62i·41-s − 0.528i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.316138050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.316138050\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 1.73iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 - 9T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 13.8iT - 67T^{2} \) |
| 71 | \( 1 + 13.8iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 5.19iT - 79T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 - 3.46iT - 89T^{2} \) |
| 97 | \( 1 + 19.0iT - 97T^{2} \) |
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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
−8.704830156572790760246888264149, −8.209241408384576482995906855853, −7.23190518295011861673334140907, −6.36161622278301603087279600668, −5.65312826915049150368953992614, −4.89857073254496919567266511944, −4.11590767657036919157033494100, −3.05483030106941736035849191117, −1.65742590566395821556650286693, −0.58037866092705582547067216376,
1.09663206431121399173220242872, 2.51038620462384590167736328847, 3.30416095163050587340862252201, 4.55367243441188145791000733738, 5.09521081815819592401997890501, 6.26986701566358478026666676371, 6.69254799408321780320291743569, 7.52362276495198474154645649735, 8.221151035289677986286238852069, 9.435205373578218656466347563017
