| L(s) = 1 | + 4.13i·5-s − 2.64·7-s − 17.3i·11-s + 1.76·13-s + 9.02i·17-s − 27.3·19-s − 8.31i·23-s + 7.87·25-s − 35.4i·29-s − 45.7·31-s − 10.9i·35-s + 65.0·37-s − 55.5i·41-s + 46.5·43-s − 37.4i·47-s + ⋯ |
| L(s) = 1 | + 0.827i·5-s − 0.377·7-s − 1.57i·11-s + 0.135·13-s + 0.530i·17-s − 1.44·19-s − 0.361i·23-s + 0.315·25-s − 1.22i·29-s − 1.47·31-s − 0.312i·35-s + 1.75·37-s − 1.35i·41-s + 1.08·43-s − 0.797i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.068416551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.068416551\) |
| \(L(2)\) |
|
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 | \( 1 + 2.64T \) |
| good | 5 | \( 1 - 4.13iT - 25T^{2} \) |
| 11 | \( 1 + 17.3iT - 121T^{2} \) |
| 13 | \( 1 - 1.76T + 169T^{2} \) |
| 17 | \( 1 - 9.02iT - 289T^{2} \) |
| 19 | \( 1 + 27.3T + 361T^{2} \) |
| 23 | \( 1 + 8.31iT - 529T^{2} \) |
| 29 | \( 1 + 35.4iT - 841T^{2} \) |
| 31 | \( 1 + 45.7T + 961T^{2} \) |
| 37 | \( 1 - 65.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 55.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 46.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 37.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 0.0153iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 88.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 1.01T + 3.72e3T^{2} \) |
| 67 | \( 1 + 23.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 69.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 100.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 64.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 56.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 39.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 19.6T + 9.40e3T^{2} \) |
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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
−10.08139798039818906533408051854, −8.967677925541079648971204970315, −8.311191257131522536042066989238, −7.26286752183436280477895330125, −6.24214966904001972873372239339, −5.82800971256996952951454049716, −4.21307904910398538176848788259, −3.32741651735493205892368488316, −2.26864987834278138947924095886, −0.37654096643183380859752236100,
1.35846107938571754513758883370, 2.63101307428717097411379296911, 4.16429368481307298216893640326, 4.79277148857386044227469891720, 5.90015630763933589507952800121, 6.96480251724726158536746151356, 7.70503936711657017774468063423, 8.875969811385718560791351768296, 9.362791939719258623984858570420, 10.26636650817196111540391348196
