| L(s) = 1 | + 2.82i·5-s + i·7-s + 1.41·11-s − 6·13-s + 5.65i·17-s − 8i·19-s − 1.41·23-s − 3.00·25-s − 4.24i·29-s + 4i·31-s − 2.82·35-s − 8·37-s − 8.48i·41-s + 12i·43-s − 2.82·47-s + ⋯ |
| L(s) = 1 | + 1.26i·5-s + 0.377i·7-s + 0.426·11-s − 1.66·13-s + 1.37i·17-s − 1.83i·19-s − 0.294·23-s − 0.600·25-s − 0.787i·29-s + 0.718i·31-s − 0.478·35-s − 1.31·37-s − 1.32i·41-s + 1.82i·43-s − 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 - 5.65iT - 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 8.48iT - 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 1.41iT - 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 10iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 14.1iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.081911224584074154301695266775, −7.33286819098264479571518605039, −6.72703754146527397301644079437, −6.20099601355347731446877621802, −5.15109049371457831039649020513, −4.43129945513003024825216650134, −3.34038358411336735405385424237, −2.66243303672476388568699677906, −1.86191165156284044426051644246, 0,
1.19535508099653498395602079920, 2.16915247958431223576993142423, 3.37965203909395327447968569863, 4.27542148405272055962733025667, 5.01782713062812631051384436736, 5.45246354739394560912715681948, 6.56838046709271704416192446899, 7.37296149750019489915511236238, 7.924553684666347829684150489498
