| L(s) = 1 | + (−1.70 + 0.306i)3-s + 2.09i·5-s + 0.613i·7-s + (2.81 − 1.04i)9-s − 13-s + (−0.641 − 3.56i)15-s + 2.09i·17-s + 5.29i·19-s + (−0.188 − 1.04i)21-s − 6.81·23-s + 0.623·25-s + (−4.47 + 2.64i)27-s − 6.92i·29-s + 5.29i·31-s − 1.28·35-s + ⋯ |
| L(s) = 1 | + (−0.984 + 0.177i)3-s + 0.935i·5-s + 0.231i·7-s + (0.937 − 0.348i)9-s − 0.277·13-s + (−0.165 − 0.920i)15-s + 0.507i·17-s + 1.21i·19-s + (−0.0410 − 0.228i)21-s − 1.42·23-s + 0.124·25-s + (−0.860 + 0.509i)27-s − 1.28i·29-s + 0.950i·31-s − 0.216·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4873901370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4873901370\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 - 0.306i)T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2.09iT - 5T^{2} \) |
| 7 | \( 1 - 0.613iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 2.09iT - 17T^{2} \) |
| 19 | \( 1 - 5.29iT - 19T^{2} \) |
| 23 | \( 1 + 6.81T + 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 5.29iT - 31T^{2} \) |
| 37 | \( 1 - 5.62T + 37T^{2} \) |
| 41 | \( 1 - 11.1iT - 41T^{2} \) |
| 43 | \( 1 + 11.1iT - 43T^{2} \) |
| 47 | \( 1 + 3.40T + 47T^{2} \) |
| 53 | \( 1 - 11.1iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 5.29iT - 67T^{2} \) |
| 71 | \( 1 + 3.40T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 6.51iT - 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 2.74iT - 89T^{2} \) |
| 97 | \( 1 + 5.24T + 97T^{2} \) |
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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
−9.629536933769724841977958357304, −8.470971378387610825858327232378, −7.65300293373059579509226079280, −6.92457184917910781303686861434, −6.01107551380745498442964432979, −5.79410722817141449011501257620, −4.51104008892979280826897174346, −3.84211717083580245458118798543, −2.70750064749833897144164178019, −1.52300679775488622018422461980,
0.20200117911358889412527929963, 1.22711197543828952406694072727, 2.45128899987855199351730413423, 3.94429907772012776668080823201, 4.71791632600789254846572605012, 5.28052060074180457682256785356, 6.12442880831850542935476000113, 6.97555190739949457672749507876, 7.62542571276722324514801991185, 8.528377411570988977393638058629
