| L(s) = 1 | + 4.29i·5-s + 7-s − 1.60i·11-s − 6.02i·13-s − 3.16·17-s + 3.07i·19-s + 5.95·23-s − 13.4·25-s − 3.53i·29-s + 2.08·31-s + 4.29i·35-s − 4.48i·37-s + 6.69·41-s − 12.2i·43-s + 3.82·47-s + ⋯ |
| L(s) = 1 | + 1.92i·5-s + 0.377·7-s − 0.484i·11-s − 1.67i·13-s − 0.766·17-s + 0.706i·19-s + 1.24·23-s − 2.69·25-s − 0.657i·29-s + 0.375·31-s + 0.726i·35-s − 0.736i·37-s + 1.04·41-s − 1.87i·43-s + 0.558·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.747756961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.747756961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 4.29iT - 5T^{2} \) |
| 11 | \( 1 + 1.60iT - 11T^{2} \) |
| 13 | \( 1 + 6.02iT - 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 3.07iT - 19T^{2} \) |
| 23 | \( 1 - 5.95T + 23T^{2} \) |
| 29 | \( 1 + 3.53iT - 29T^{2} \) |
| 31 | \( 1 - 2.08T + 31T^{2} \) |
| 37 | \( 1 + 4.48iT - 37T^{2} \) |
| 41 | \( 1 - 6.69T + 41T^{2} \) |
| 43 | \( 1 + 12.2iT - 43T^{2} \) |
| 47 | \( 1 - 3.82T + 47T^{2} \) |
| 53 | \( 1 + 8.63iT - 53T^{2} \) |
| 59 | \( 1 - 1.55iT - 59T^{2} \) |
| 61 | \( 1 + 3.64iT - 61T^{2} \) |
| 67 | \( 1 - 0.772iT - 67T^{2} \) |
| 71 | \( 1 - 7.36T + 71T^{2} \) |
| 73 | \( 1 - 1.32T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 7.08iT - 83T^{2} \) |
| 89 | \( 1 + 9.73T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.939018605103857466007195887694, −7.25818124249134534791853655663, −6.75657798520392570730256764894, −5.84346088463692107062241567527, −5.48848725694052251013446437042, −4.19928357853064438021837591334, −3.40771541561357110877065369316, −2.81888120562780622085077660121, −2.08097445481705132559141053669, −0.50561342925499237006933202823,
1.00693515594459741845397051248, 1.64638275559828341836632454807, 2.65177789434095954135858929163, 4.12492958717606087367847774231, 4.58602908438366528167497845082, 4.92434651315496129158260694135, 5.88147527398660277793833236818, 6.76288907479535268602705572480, 7.41397389186502773834208036920, 8.341694812441024951474562019470
