| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.308112383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.308112383\) |
| \(L(\frac{3}{2})\) |
|
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
−9.001459039225397070308812829286, −8.212846295492343883968322783125, −7.58494722152220269601368204874, −6.65030428684729264818970089276, −6.17343778876993231085065776638, −4.92270996042427055340456072908, −4.14301318316900065635135627308, −2.92619016772606804013986592059, −2.64127991746733602747381643297, −1.01962032116672229306102701652,
0.962526604874529470469330704873, 2.10302338141850890160574972475, 3.09280100643736370452388762144, 4.31648877412164219233067957796, 4.66511463529891051653281781832, 5.86702366733265572321818160965, 6.62233196555210255726968134397, 7.59730443176789810781247623848, 8.282087060702713190370508050758, 8.852419166544719296225306386758
