| L(s) = 1 | + 0.306i·3-s − 2.60i·7-s + 2.90·9-s + 2.71·11-s − 0.503i·13-s + 5.94i·17-s + 5.72·19-s + 0.799·21-s − 1.28i·23-s + 1.81i·27-s − 2.11·29-s + 3.95·31-s + 0.831i·33-s + 0.825i·37-s + 0.154·39-s + ⋯ |
| L(s) = 1 | + 0.176i·3-s − 0.985i·7-s + 0.968·9-s + 0.818·11-s − 0.139i·13-s + 1.44i·17-s + 1.31·19-s + 0.174·21-s − 0.268i·23-s + 0.348i·27-s − 0.392·29-s + 0.710·31-s + 0.144i·33-s + 0.135i·37-s + 0.0246·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.341334368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.341334368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.306iT - 3T^{2} \) |
| 7 | \( 1 + 2.60iT - 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 + 0.503iT - 13T^{2} \) |
| 17 | \( 1 - 5.94iT - 17T^{2} \) |
| 19 | \( 1 - 5.72T + 19T^{2} \) |
| 23 | \( 1 + 1.28iT - 23T^{2} \) |
| 29 | \( 1 + 2.11T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 - 0.825iT - 37T^{2} \) |
| 41 | \( 1 + 4.53T + 41T^{2} \) |
| 43 | \( 1 - 5.38iT - 43T^{2} \) |
| 47 | \( 1 - 5.62iT - 47T^{2} \) |
| 53 | \( 1 + 10.9iT - 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 7.00T + 61T^{2} \) |
| 67 | \( 1 - 2.85iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 10.1iT - 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 5.83iT - 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 7.02iT - 97T^{2} \) |
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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
−8.372969162505242410560318921969, −7.67164165729430461276154082386, −6.97639971187206771505235934338, −6.40396303678978662247444718085, −5.44473799146764295973591890567, −4.44928356940705091490909803987, −3.95298573379079286218799589903, −3.19689926592584631656563085779, −1.71107400391760168829292155130, −0.974295861166339251123190171086,
0.915256463997868912796970581395, 1.94516425331264553051481797436, 2.91203111632549802402398799734, 3.81524129493345317976325938946, 4.78937250188975092036891343517, 5.42842211325491932501383546643, 6.27867802716520724464260757809, 7.12983332746628057564067134505, 7.49405381665830138392612920156, 8.566918253227958342263826728344
