| L(s) = 1 | + 3-s − 3.46i·5-s + 9-s + 3.46i·11-s + (−1 − 3.46i)13-s − 3.46i·15-s + 6·17-s − 6.92i·19-s − 6.99·25-s + 27-s − 6·29-s − 6.92i·31-s + 3.46i·33-s + (−1 − 3.46i)39-s + 3.46i·41-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.54i·5-s + 0.333·9-s + 1.04i·11-s + (−0.277 − 0.960i)13-s − 0.894i·15-s + 1.45·17-s − 1.58i·19-s − 1.39·25-s + 0.192·27-s − 1.11·29-s − 1.24i·31-s + 0.603i·33-s + (−0.160 − 0.554i)39-s + 0.541i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39805 - 1.05155i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39805 - 1.05155i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 6.92iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 3.46iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 13.8iT - 67T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 17.3iT - 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−10.03319919006097690409661422494, −9.571121095460285250558534259341, −8.695462472836187073491672821735, −7.899157899631497257729537097740, −7.16381086658599270667006327285, −5.55903870820136015258424800715, −4.91253698194720038345851672365, −3.87971968651656524035706044867, −2.42756358190572304351237548972, −0.962378237056993246340949994429,
1.89274830270337970362840279571, 3.25340772811338126440654314627, 3.70510253596749055525576898377, 5.48434406324512321319558983858, 6.41993933835493385426694212487, 7.29920581887175946111135754089, 8.033184421989038726705712112602, 9.073372844835513293847009131565, 10.11896025219599300538723330682, 10.56444373348329980395526127698
