L(s) = 1 | + (−2.41 − 2.41i)5-s + 1.53i·7-s + (−3.37 − 3.37i)11-s + (0.414 − 0.414i)13-s − 2.82·17-s + (−0.317 + 0.317i)19-s + 5.86i·23-s + 6.65i·25-s + (3.24 − 3.24i)29-s − 7.39·31-s + (3.69 − 3.69i)35-s + (−3.58 − 3.58i)37-s − 4i·41-s + (1.84 + 1.84i)43-s − 7.39·47-s + ⋯ |
L(s) = 1 | + (−1.07 − 1.07i)5-s + 0.578i·7-s + (−1.01 − 1.01i)11-s + (0.114 − 0.114i)13-s − 0.685·17-s + (−0.0727 + 0.0727i)19-s + 1.22i·23-s + 1.33i·25-s + (0.602 − 0.602i)29-s − 1.32·31-s + (0.624 − 0.624i)35-s + (−0.589 − 0.589i)37-s − 0.624i·41-s + (0.281 + 0.281i)43-s − 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6554507069\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6554507069\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.41 + 2.41i)T + 5iT^{2} \) |
| 7 | \( 1 - 1.53iT - 7T^{2} \) |
| 11 | \( 1 + (3.37 + 3.37i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.414 + 0.414i)T - 13iT^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + (0.317 - 0.317i)T - 19iT^{2} \) |
| 23 | \( 1 - 5.86iT - 23T^{2} \) |
| 29 | \( 1 + (-3.24 + 3.24i)T - 29iT^{2} \) |
| 31 | \( 1 + 7.39T + 31T^{2} \) |
| 37 | \( 1 + (3.58 + 3.58i)T + 37iT^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + (-1.84 - 1.84i)T + 43iT^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 + (-5.24 - 5.24i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.84 + 1.84i)T + 59iT^{2} \) |
| 61 | \( 1 + (9.24 - 9.24i)T - 61iT^{2} \) |
| 67 | \( 1 + (-7.07 + 7.07i)T - 67iT^{2} \) |
| 71 | \( 1 - 11.9iT - 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 6.12T + 79T^{2} \) |
| 83 | \( 1 + (2.48 - 2.48i)T - 83iT^{2} \) |
| 89 | \( 1 - 0.828iT - 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.463591105651719270877662401003, −7.80196202497893155171266900341, −7.21658207044421090669498309873, −5.97285452993048575351200821782, −5.46037998031720625678328968352, −4.74217933750587342318560234529, −3.87800028003871213066280759878, −3.15158620276726373268487186588, −2.03015138138485603592960724758, −0.71242652299642849485265853637,
0.26445304805312311599892229862, 1.97481990605412878085470046784, 2.86427956087731497967436309985, 3.64691663000612211618027301337, 4.45749368943716046423634438739, 5.05030329360642257244201159238, 6.33049529007143796565314603562, 6.98209068763786306034997309729, 7.35633348917798185807975576495, 8.092751011425633119754834713548