| L(s) = 1 | − 2.17i·2-s − 2.70·4-s + (−0.539 − 2.17i)5-s + 0.829i·7-s + 1.53i·8-s + (−4.70 + 1.17i)10-s − 2.53·11-s − 4.24i·13-s + 1.80·14-s − 2.07·16-s − 1.36i·17-s + 19-s + (1.46 + 5.87i)20-s + 5.51i·22-s + 3.36i·23-s + ⋯ |
| L(s) = 1 | − 1.53i·2-s − 1.35·4-s + (−0.241 − 0.970i)5-s + 0.313i·7-s + 0.544i·8-s + (−1.48 + 0.370i)10-s − 0.765·11-s − 1.17i·13-s + 0.481·14-s − 0.519·16-s − 0.332i·17-s + 0.229·19-s + (0.326 + 1.31i)20-s + 1.17i·22-s + 0.702i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.460979 + 0.589532i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.460979 + 0.589532i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.539 + 2.17i)T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.17iT - 2T^{2} \) |
| 7 | \( 1 - 0.829iT - 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 1.36iT - 17T^{2} \) |
| 23 | \( 1 - 3.36iT - 23T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 + 0.290T + 31T^{2} \) |
| 37 | \( 1 - 7.51iT - 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 5.35iT - 43T^{2} \) |
| 47 | \( 1 + 8.63iT - 47T^{2} \) |
| 53 | \( 1 + 1.86iT - 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 7.86T + 61T^{2} \) |
| 67 | \( 1 - 4.15iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 3.57iT - 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 18.6iT - 97T^{2} \) |
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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
−9.903899566563886359874245830605, −8.896221621498591821590287068170, −8.255663292772664587241450360679, −7.21885579469546472329711518347, −5.55506393203565031723350303441, −4.98899736065650528083720281913, −3.78479322829486426715792855984, −2.90801888464049000252612705113, −1.68751352485136195328152211760, −0.35377050200424766631527593355,
2.32713486116535707865553946972, 3.79108086986958228401019678199, 4.76660788682448929702652295339, 5.85091840056548609142121893457, 6.61278123321554928916578731716, 7.27868331890556514071462813825, 7.905080678006886327504594273841, 8.820302486622486628805183804998, 9.751319578100703223195205919102, 10.75136957906575420907214222936
