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
−8.092751011425633119754834713548, −7.35633348917798185807975576495, −6.98209068763786306034997309729, −6.33049529007143796565314603562, −5.05030329360642257244201159238, −4.45749368943716046423634438739, −3.64691663000612211618027301337, −2.86427956087731497967436309985, −1.97481990605412878085470046784, −0.26445304805312311599892229862,
0.71242652299642849485265853637, 2.03015138138485603592960724758, 3.15158620276726373268487186588, 3.87800028003871213066280759878, 4.74217933750587342318560234529, 5.46037998031720625678328968352, 5.97285452993048575351200821782, 7.21658207044421090669498309873, 7.80196202497893155171266900341, 8.463591105651719270877662401003