| L(s) = 1 | + 2-s + (−1 + i)3-s + 4-s − i·5-s + (−1 + i)6-s − i·7-s + 8-s − i·9-s − i·10-s + (−1 + i)12-s + 13-s − i·14-s + (1 + i)15-s + 16-s − i·18-s + (−1 − i)19-s + ⋯ |
| L(s) = 1 | + 2-s + (−1 + i)3-s + 4-s − i·5-s + (−1 + i)6-s − i·7-s + 8-s − i·9-s − i·10-s + (−1 + i)12-s + 13-s − i·14-s + (1 + i)15-s + 16-s − i·18-s + (−1 − i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.714852966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.714852966\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + (1 - i)T - iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (1 + i)T + iT^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-1 + i)T - iT^{2} \) |
| 61 | \( 1 - 2T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-1 + i)T - iT^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.544345056445976349492499104785, −7.907665635195111643330277210225, −6.68509782513659686960183927936, −6.28985004953035691290003170946, −5.33078143950019416599560203900, −4.84626641013200882682640975605, −4.01941460949354785046818925767, −3.85166519534411466326304059417, −2.20727636591982614509305321123, −0.812027250758544947687830009882,
1.61683073770384663897155076520, 2.26305586879485825946469400948, 3.38183778792545703963324931736, 4.13735235320221606110913384325, 5.59771954005026833664794158358, 5.72790987087233118687894375569, 6.43110718057658224877387082645, 6.93685999616736560998139986668, 7.81054220311668269357792988974, 8.446336910265357698826000808240
