| L(s) = 1 | + (−1.71 + 0.238i)3-s − 1.32·5-s + (2.88 − 0.817i)9-s + 2.81i·11-s − 2.52i·13-s + (2.27 − 0.315i)15-s − 3.26·17-s − 5.73i·19-s + 4.25i·23-s − 3.24·25-s + (−4.75 + 2.09i)27-s − 4.75i·29-s + 9.10i·31-s + (−0.669 − 4.82i)33-s + 11.2·37-s + ⋯ |
| L(s) = 1 | + (−0.990 + 0.137i)3-s − 0.592·5-s + (0.962 − 0.272i)9-s + 0.847i·11-s − 0.701i·13-s + (0.586 − 0.0814i)15-s − 0.791·17-s − 1.31i·19-s + 0.887i·23-s − 0.649·25-s + (−0.915 + 0.402i)27-s − 0.883i·29-s + 1.63i·31-s + (−0.116 − 0.839i)33-s + 1.85·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9140120533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9140120533\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.71 - 0.238i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.32T + 5T^{2} \) |
| 11 | \( 1 - 2.81iT - 11T^{2} \) |
| 13 | \( 1 + 2.52iT - 13T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 19 | \( 1 + 5.73iT - 19T^{2} \) |
| 23 | \( 1 - 4.25iT - 23T^{2} \) |
| 29 | \( 1 + 4.75iT - 29T^{2} \) |
| 31 | \( 1 - 9.10iT - 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.48T + 43T^{2} \) |
| 47 | \( 1 - 1.48T + 47T^{2} \) |
| 53 | \( 1 + 12.0iT - 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 2.34iT - 61T^{2} \) |
| 67 | \( 1 + 0.130T + 67T^{2} \) |
| 71 | \( 1 + 0.875iT - 71T^{2} \) |
| 73 | \( 1 - 6.44iT - 73T^{2} \) |
| 79 | \( 1 - 5.93T + 79T^{2} \) |
| 83 | \( 1 - 0.398T + 83T^{2} \) |
| 89 | \( 1 - 5.75T + 89T^{2} \) |
| 97 | \( 1 - 5.36iT - 97T^{2} \) |
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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
−9.807644562354125600817316239547, −9.159397384229574386183617707928, −7.908640817768437143891112278167, −7.25548640435591456604344572515, −6.45041943842887168374884231812, −5.46195627166687257747352733039, −4.62217059142636069854772080434, −3.88251931238977011338225575825, −2.39816793829829994235577234533, −0.71902527057228496579709286989,
0.78159667046043304798219968448, 2.32783556444908572230772002586, 3.97335960721707266758791757401, 4.42519661309161903084497547046, 5.86113915862576239351591359829, 6.15685615169358626229872745914, 7.34030486954332510951948370672, 7.948066124108878459821109697521, 8.996732865551130355717221531774, 9.852401877520073567717462877171
