| L(s) = 1 | + (−2 − i)5-s − 5i·7-s + 4·11-s − 5i·13-s + 7·19-s − i·23-s + (3 + 4i)25-s + 8·29-s + 6·31-s + (−5 + 10i)35-s + 2i·37-s + 41-s − 6i·43-s + 3i·47-s − 18·49-s + ⋯ |
| L(s) = 1 | + (−0.894 − 0.447i)5-s − 1.88i·7-s + 1.20·11-s − 1.38i·13-s + 1.60·19-s − 0.208i·23-s + (0.600 + 0.800i)25-s + 1.48·29-s + 1.07·31-s + (−0.845 + 1.69i)35-s + 0.328i·37-s + 0.156·41-s − 0.914i·43-s + 0.437i·47-s − 2.57·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.773793673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.773793673\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| good | 7 | \( 1 + 5iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + 5iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.122363914215625404509134907329, −7.77579063357639884124368843227, −7.02511229787380944644404450095, −6.37100990275788189639824573582, −5.09748967704765382424665637500, −4.48572990879663811298662254339, −3.65594157796337916906888451346, −3.14350044318165741166100186621, −1.11993476585825243543626752281, −0.72551168355155893685546516186,
1.32487325532347433708638408481, 2.53587172005214100289893803273, 3.23988083663932460683747761151, 4.26461682711633204992295064241, 4.99346641996509709966668455258, 6.07702857582457450779528451176, 6.54902217043252638088100068813, 7.38033383980527697006230033056, 8.276762096577964556977062174990, 8.942849650722421386068305288317
