| L(s) = 1 | + i·2-s − 4-s + 3.86i·5-s + (1.61 − 2.09i)7-s − i·8-s − 3.86·10-s − 1.10·11-s + 5.86·13-s + (2.09 + 1.61i)14-s + 16-s − 5.87i·17-s + (0.266 − 4.35i)19-s − 3.86i·20-s − 1.10i·22-s + 7.40·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.72i·5-s + (0.609 − 0.792i)7-s − 0.353i·8-s − 1.22·10-s − 0.334·11-s + 1.62·13-s + (0.560 + 0.431i)14-s + 0.250·16-s − 1.42i·17-s + (0.0611 − 0.998i)19-s − 0.864i·20-s − 0.236i·22-s + 1.54·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.970519225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.970519225\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.61 + 2.09i)T \) |
| 19 | \( 1 + (-0.266 + 4.35i)T \) |
| good | 5 | \( 1 - 3.86iT - 5T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 - 5.86T + 13T^{2} \) |
| 17 | \( 1 + 5.87iT - 17T^{2} \) |
| 23 | \( 1 - 7.40T + 23T^{2} \) |
| 29 | \( 1 + 3.97iT - 29T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 + 2.18iT - 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 1.61T + 43T^{2} \) |
| 47 | \( 1 + 9.52iT - 47T^{2} \) |
| 53 | \( 1 + 3.47iT - 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 7.08iT - 61T^{2} \) |
| 67 | \( 1 - 4.38iT - 67T^{2} \) |
| 71 | \( 1 - 6.72iT - 71T^{2} \) |
| 73 | \( 1 + 3.51iT - 73T^{2} \) |
| 79 | \( 1 - 14.8iT - 79T^{2} \) |
| 83 | \( 1 - 1.08iT - 83T^{2} \) |
| 89 | \( 1 + 4.98T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−8.987712143755701305692917310912, −8.055385406257163670758719259050, −7.30676966009390078955359512560, −6.91581812600798807404783429157, −6.20096872817474221555493959692, −5.23263683108964723430042372412, −4.27451338720457228810378049063, −3.35390148576155843921573372387, −2.54606362659853504144426262778, −0.826458720896173732737487348162,
1.20571393337003449168409025362, 1.57780241906127022435916940398, 3.04772615543383805022619192088, 4.14561320773194997903056450943, 4.73078604097499536514112874777, 5.68037102408337098198719005518, 6.07446375590217092599647497087, 7.87457545148657385771645439252, 8.251896603318525867302985744311, 9.030694836003182328743658698196
