| L(s) = 1 | − 1.41i·5-s − 1.41·7-s − 2i·11-s − 2·17-s − 4i·19-s + 2.82·23-s + 2.99·25-s − 9.89i·29-s − 7.07·31-s + 2.00i·35-s + 8.48i·37-s + 6·41-s + 8i·43-s − 2.82·47-s − 5·49-s + ⋯ |
| L(s) = 1 | − 0.632i·5-s − 0.534·7-s − 0.603i·11-s − 0.485·17-s − 0.917i·19-s + 0.589·23-s + 0.599·25-s − 1.83i·29-s − 1.27·31-s + 0.338i·35-s + 1.39i·37-s + 0.937·41-s + 1.21i·43-s − 0.412·47-s − 0.714·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5811132559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5811132559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 9.89iT - 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 1.41iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 - 14.1iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.112020267408280086958281863239, −7.18151052660050347403549247818, −6.47136926321576403607778829587, −5.81581248942971836272300801721, −4.90500264392061985260636658766, −4.31783802286267246989101793499, −3.26906844975400471772941219146, −2.53689249433176150367220977743, −1.24496234369651940898923662399, −0.16411757004543384290744746077,
1.47752378891968983728573047630, 2.49135906024492660545991812208, 3.35310689775266721256322981158, 4.03859908282393807157700676854, 5.08133374374905111586069591822, 5.76431547908572845294945901719, 6.71103097724911089403601970632, 7.10563297383326096644860155814, 7.78519246012904452542030545318, 8.915506404428745398050616273280
