| L(s) = 1 | + (2.64 − 1.41i)3-s − 0.913i·5-s + 2.64·7-s + (5 − 7.48i)9-s + 14.5i·11-s + 0.583·13-s + (−1.29 − 2.41i)15-s + 21.5i·17-s − 16·19-s + (7.00 − 3.74i)21-s + 38.8i·23-s + 24.1·25-s + (2.64 − 26.8i)27-s + 35.7i·29-s + 58.4·31-s + ⋯ |
| L(s) = 1 | + (0.881 − 0.471i)3-s − 0.182i·5-s + 0.377·7-s + (0.555 − 0.831i)9-s + 1.32i·11-s + 0.0448·13-s + (−0.0861 − 0.161i)15-s + 1.26i·17-s − 0.842·19-s + (0.333 − 0.178i)21-s + 1.68i·23-s + 0.966·25-s + (0.0979 − 0.995i)27-s + 1.23i·29-s + 1.88·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.799775145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.799775145\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.64 + 1.41i)T \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 + 0.913iT - 25T^{2} \) |
| 11 | \( 1 - 14.5iT - 121T^{2} \) |
| 13 | \( 1 - 0.583T + 169T^{2} \) |
| 17 | \( 1 - 21.5iT - 289T^{2} \) |
| 19 | \( 1 + 16T + 361T^{2} \) |
| 23 | \( 1 - 38.8iT - 529T^{2} \) |
| 29 | \( 1 - 35.7iT - 841T^{2} \) |
| 31 | \( 1 - 58.4T + 961T^{2} \) |
| 37 | \( 1 + 20T + 1.36e3T^{2} \) |
| 41 | \( 1 - 8.75iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 11.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 8.48iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 50.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 58.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 38.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 70.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 17.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 72.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 20.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 145. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 53.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 111.T + 9.40e3T^{2} \) |
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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
−9.436353757203469808795150648329, −8.488148384707539334179057687418, −8.048750861312848830879170777800, −7.05923364889935444729977752528, −6.48852920797660142842569397562, −5.16318364153741727954087401821, −4.26985728351885698000024273931, −3.32600193050865107546479930559, −2.07828270395232345718808603059, −1.34670068609124476607195960970,
0.75080687751282706412307591182, 2.43949638614689092292260292371, 3.02148414376764412602196133756, 4.24265750834971279965417268710, 4.88518116354177266035271342333, 6.10532842747650649085549452549, 6.95877504202707445564459996082, 8.068993315978860103143965814465, 8.508400544458299707812958242754, 9.197910202544974310961958143780
