L(s) = 1 | + (−0.707 + 2.12i)5-s + (2 + 2i)7-s + 2.82i·11-s + (−3 + 3i)13-s + (−1.41 + 1.41i)17-s − 4i·19-s + (5.65 + 5.65i)23-s + (−3.99 − 3i)25-s + 9.89·29-s − 8·31-s + (−5.65 + 2.82i)35-s + (3 + 3i)37-s + 1.41i·41-s + (−8.48 + 8.48i)47-s + i·49-s + ⋯ |
L(s) = 1 | + (−0.316 + 0.948i)5-s + (0.755 + 0.755i)7-s + 0.852i·11-s + (−0.832 + 0.832i)13-s + (−0.342 + 0.342i)17-s − 0.917i·19-s + (1.17 + 1.17i)23-s + (−0.799 − 0.600i)25-s + 1.83·29-s − 1.43·31-s + (−0.956 + 0.478i)35-s + (0.493 + 0.493i)37-s + 0.220i·41-s + (−1.23 + 1.23i)47-s + 0.142i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.290159037\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.290159037\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 2.12i)T \) |
good | 7 | \( 1 + (-2 - 2i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.41 - 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (-5.65 - 5.65i)T + 23iT^{2} \) |
| 29 | \( 1 - 9.89T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + (8.48 - 8.48i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.07 + 7.07i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + (8 + 8i)T + 67iT^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (-7 + 7i)T - 73iT^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 1.41T + 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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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
−9.265759951826555101367123520135, −8.246381462122685831113856715314, −7.53388063691546649417273300808, −6.87990276555014877872152810534, −6.25873980970075290635018565099, −4.92665121799502898290279585845, −4.70570195082917290157367408838, −3.38350120180568972042326715239, −2.50369773389795496160012895247, −1.69596936132741037757121597360,
0.42169638626804212821357084229, 1.32746091118411572805236413950, 2.71163726827378816568677018601, 3.74603047936285064636462811879, 4.68102624445576905568846525337, 5.10578216836093293154736559604, 6.06247634840820631062050330689, 7.12448002307452438694499486475, 7.79815913638830886054427863618, 8.414226746929542733216971778370