| L(s) = 1 | − 3-s − 1.73i·5-s − 2·9-s − 1.73i·11-s + 1.73i·15-s + 5.19i·17-s − 7·19-s + 8.66i·23-s + 2.00·25-s + 5·27-s + 6·29-s + 5·31-s + 1.73i·33-s + 5·37-s − 6.92i·41-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.774i·5-s − 0.666·9-s − 0.522i·11-s + 0.447i·15-s + 1.26i·17-s − 1.60·19-s + 1.80i·23-s + 0.400·25-s + 0.962·27-s + 1.11·29-s + 0.898·31-s + 0.301i·33-s + 0.821·37-s − 1.08i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.154980669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.154980669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 1.73iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 - 8.66iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 + 8.66iT - 61T^{2} \) |
| 67 | \( 1 - 5.19iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 + 5.19iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 12.1iT - 89T^{2} \) |
| 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
−8.536428923651488807045918370325, −8.136834600189938614514045555944, −6.98781947117146771214434162195, −6.06830755076197169956216885626, −5.72989050044370266438141976182, −4.77816580377407432374688501531, −4.03709187508633558527002261586, −2.98877441346360831430960445748, −1.76310070209963987818281819017, −0.58600254898219169023906617950,
0.74451300334969369936724884526, 2.55818084243568342478986689104, 2.78938344211452363029074565718, 4.36755663456722172677625188865, 4.77543365454055639206429654275, 5.92695331126394683189090261644, 6.57260244251443030947066863262, 6.96433645252916185442211899555, 8.128745841724463020465797181739, 8.641668440583450616374179560073
