L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 4·7-s − 0.999·8-s + (1.5 + 2.59i)9-s − 11-s + (1 + 1.73i)13-s + (−2 + 3.46i)14-s + (−0.5 + 0.866i)16-s + (−1.5 + 2.59i)17-s + 3·18-s + (4 + 1.73i)19-s + (−0.5 + 0.866i)22-s + (2 + 3.46i)23-s + 1.99·26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 1.51·7-s − 0.353·8-s + (0.5 + 0.866i)9-s − 0.301·11-s + (0.277 + 0.480i)13-s + (−0.534 + 0.925i)14-s + (−0.125 + 0.216i)16-s + (−0.363 + 0.630i)17-s + 0.707·18-s + (0.917 + 0.397i)19-s + (−0.106 + 0.184i)22-s + (0.417 + 0.722i)23-s + 0.392·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07510 + 0.487489i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07510 + 0.487489i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-4 - 1.73i)T \) |
good | 3 | \( 1 + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6 - 10.3i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.5 + 12.9i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.16859351086247851025105791810, −9.576480149181028029468024761089, −8.706108431812127578304744353087, −7.52890313441364507043717555437, −6.66311011722927919707263303019, −5.76470860108653836015411529248, −4.77448897721646626211453931696, −3.68362201472377614093225082887, −2.87848973299083312316921824190, −1.52807265628850812338325120329,
0.50100639163660691663184068019, 2.82940459869884827336252932759, 3.54979302518516662793457994536, 4.66295310994620697106268586990, 5.77477784657855480082064813370, 6.58024055513476895465898157119, 7.08551808863849815405925565338, 8.159009584040943836754208767695, 9.271726308994816244280619594904, 9.640071709600070347331925838773