| L(s) = 1 | + 3-s − 5-s + 3.67i·7-s + 9-s + 4.75i·11-s + 4.23i·13-s − 15-s + 1.85·17-s + (1.32 − 4.15i)19-s + 3.67i·21-s − 4.55i·23-s + 25-s + 27-s + 4.50i·29-s + 2.95·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.38i·7-s + 0.333·9-s + 1.43i·11-s + 1.17i·13-s − 0.258·15-s + 0.450·17-s + (0.302 − 0.952i)19-s + 0.801i·21-s − 0.950i·23-s + 0.200·25-s + 0.192·27-s + 0.836i·29-s + 0.531·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.673 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.828790499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.828790499\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (-1.32 + 4.15i)T \) |
| good | 7 | \( 1 - 3.67iT - 7T^{2} \) |
| 11 | \( 1 - 4.75iT - 11T^{2} \) |
| 13 | \( 1 - 4.23iT - 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 23 | \( 1 + 4.55iT - 23T^{2} \) |
| 29 | \( 1 - 4.50iT - 29T^{2} \) |
| 31 | \( 1 - 2.95T + 31T^{2} \) |
| 37 | \( 1 + 2.49iT - 37T^{2} \) |
| 41 | \( 1 - 4.10iT - 41T^{2} \) |
| 43 | \( 1 + 0.970iT - 43T^{2} \) |
| 47 | \( 1 - 4.14iT - 47T^{2} \) |
| 53 | \( 1 - 14.2iT - 53T^{2} \) |
| 59 | \( 1 + 0.965T + 59T^{2} \) |
| 61 | \( 1 - 3.72T + 61T^{2} \) |
| 67 | \( 1 + 4.81T + 67T^{2} \) |
| 71 | \( 1 - 4.62T + 71T^{2} \) |
| 73 | \( 1 + 0.257T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 1.93iT - 83T^{2} \) |
| 89 | \( 1 + 2.54iT - 89T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
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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
−8.748509900795600122460534481626, −7.907546650968484161238990103386, −7.14157750507284598225210674046, −6.61999376927762329130011301263, −5.61693085822504272001848802713, −4.66625771397966351237770700599, −4.28716963691768408252526637896, −2.98287261711672167290694976636, −2.42322523791501265416010325924, −1.49483383109057590320590985703,
0.48715094650823104774841746700, 1.32173316540877004785290967124, 2.83528489944532502471607068770, 3.63232514728564849495612222859, 3.87267762232745480213287441281, 5.12814812595002611868436163620, 5.83400414231412597065792710717, 6.75914775414444855172027304956, 7.58207364834730016296461255017, 8.044452923870223937161259924572
