| L(s) = 1 | − 2i·3-s + (2 − i)5-s + i·7-s − 9-s + 2i·13-s + (−2 − 4i)15-s + 5i·17-s + 8·19-s + 2·21-s + i·23-s + (3 − 4i)25-s − 4i·27-s + 5·29-s + 5·31-s + (1 + 2i)35-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + (0.894 − 0.447i)5-s + 0.377i·7-s − 0.333·9-s + 0.554i·13-s + (−0.516 − 1.03i)15-s + 1.21i·17-s + 1.83·19-s + 0.436·21-s + 0.208i·23-s + (0.600 − 0.800i)25-s − 0.769i·27-s + 0.928·29-s + 0.898·31-s + (0.169 + 0.338i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.272261699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.272261699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 + 13T + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - 3iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.951532487729763002332319976483, −8.375765579340527809822076059588, −7.43975023962131211940246693187, −6.74325802401862703672458195401, −5.95137489070646669466531836892, −5.33328016647002968014205843757, −4.18722899653004125931631119656, −2.81783495171724879901785858667, −1.83492881296117993969960863091, −1.07977648410735711665803284198,
1.17306808228243964742902952309, 2.84274349368173337774974213154, 3.32978838267781671944453397696, 4.70235207395148745835275342817, 5.09448648118850537508863782361, 6.09269721317024906131007890852, 7.00935219860697231240192946526, 7.76313551498356507799038903369, 8.937412046044829089611044227641, 9.576148030410108098600644602424
