| L(s) = 1 | + i·3-s + 4.78i·7-s − 9-s + 1.95·11-s + 1.07i·13-s + 3.80i·17-s − 4.25·19-s − 4.78·21-s − 5.86i·23-s − i·27-s − 10.5·29-s − 4.31·31-s + 1.95i·33-s − 5.65i·37-s − 1.07·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.80i·7-s − 0.333·9-s + 0.589·11-s + 0.299i·13-s + 0.923i·17-s − 0.975·19-s − 1.04·21-s − 1.22i·23-s − 0.192i·27-s − 1.95·29-s − 0.775·31-s + 0.340i·33-s − 0.930i·37-s − 0.172·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1043388388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1043388388\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.78iT - 7T^{2} \) |
| 11 | \( 1 - 1.95T + 11T^{2} \) |
| 13 | \( 1 - 1.07iT - 13T^{2} \) |
| 17 | \( 1 - 3.80iT - 17T^{2} \) |
| 19 | \( 1 + 4.25T + 19T^{2} \) |
| 23 | \( 1 + 5.86iT - 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 4.31T + 31T^{2} \) |
| 37 | \( 1 + 5.65iT - 37T^{2} \) |
| 41 | \( 1 - 0.858T + 41T^{2} \) |
| 43 | \( 1 + 10.8iT - 43T^{2} \) |
| 47 | \( 1 + 3.00iT - 47T^{2} \) |
| 53 | \( 1 - 4.22iT - 53T^{2} \) |
| 59 | \( 1 - 4.77T + 59T^{2} \) |
| 61 | \( 1 - 3.63T + 61T^{2} \) |
| 67 | \( 1 + 6.93iT - 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 16.9iT - 73T^{2} \) |
| 79 | \( 1 - 6.62T + 79T^{2} \) |
| 83 | \( 1 + 2.58iT - 83T^{2} \) |
| 89 | \( 1 - 0.832T + 89T^{2} \) |
| 97 | \( 1 - 9.11iT - 97T^{2} \) |
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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
−7.904986851678708353476873316846, −6.88312780240574909073712609673, −6.09971862224137383727377690033, −5.71127211088971478867350439781, −4.95688359385561982292891527987, −4.05100581319490161457262371489, −3.47814905809631509290057666656, −2.25195397208124724303182785650, −1.95514384798287765140330418212, −0.02545836718785390804262432021,
1.05664966787453395142415412510, 1.76740550420837030815953667472, 3.01565311789095173838006280430, 3.79259547810227314386924770849, 4.35370097175860328120516968741, 5.30105335132903222348572389124, 6.09359546240795139301302894573, 6.95196432179895544668053257699, 7.25533123920694704023166882966, 7.84381129396235843195900014934
