| L(s) = 1 | + 1.73i·5-s − 4.35·7-s − 5.83i·11-s + 4.01i·13-s − 1.70·17-s − 2i·19-s − 6.64·23-s + 2.00·25-s + 4.68i·29-s − 4.86·31-s − 7.54i·35-s − 5.75i·37-s + 1.29·41-s + 3.54i·43-s + 10.6·47-s + ⋯ |
| L(s) = 1 | + 0.774i·5-s − 1.64·7-s − 1.75i·11-s + 1.11i·13-s − 0.414·17-s − 0.458i·19-s − 1.38·23-s + 0.400·25-s + 0.870i·29-s − 0.873·31-s − 1.27i·35-s − 0.945i·37-s + 0.201·41-s + 0.540i·43-s + 1.54·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.066647223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.066647223\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 + 4.35T + 7T^{2} \) |
| 11 | \( 1 + 5.83iT - 11T^{2} \) |
| 13 | \( 1 - 4.01iT - 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 6.64T + 23T^{2} \) |
| 29 | \( 1 - 4.68iT - 29T^{2} \) |
| 31 | \( 1 + 4.86T + 31T^{2} \) |
| 37 | \( 1 + 5.75iT - 37T^{2} \) |
| 41 | \( 1 - 1.29T + 41T^{2} \) |
| 43 | \( 1 - 3.54iT - 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 0.506iT - 53T^{2} \) |
| 59 | \( 1 + 1.87iT - 59T^{2} \) |
| 61 | \( 1 - 1.22iT - 61T^{2} \) |
| 67 | \( 1 + 1.57iT - 67T^{2} \) |
| 71 | \( 1 - 9.88T + 71T^{2} \) |
| 73 | \( 1 + 7.96T + 73T^{2} \) |
| 79 | \( 1 + 2.06T + 79T^{2} \) |
| 83 | \( 1 - 1.54iT - 83T^{2} \) |
| 89 | \( 1 + 3.96T + 89T^{2} \) |
| 97 | \( 1 - 6.12T + 97T^{2} \) |
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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
−8.385763547982500433660198038606, −7.20415718745081596173537233785, −6.85064452412788091732446854763, −6.04191456747402498076374454755, −5.75298335342830151130945816098, −4.33177489867575651744288754922, −3.54245498650578689561523723080, −3.05451266279573429717086222024, −2.14778648746021920099650509885, −0.54025154102522291117917850156,
0.52878247858443635999136782237, 1.91019295508183477551283082227, 2.77753920762187500984279374397, 3.79331001392619542424693735865, 4.39260507715414555621598068635, 5.34273513579487179640662064590, 6.00230664607471514213368160848, 6.75738338302647997093070918451, 7.45166681703746336774005176934, 8.133297886027577547629131946314
