| L(s) = 1 | + (0.707 + 2.12i)5-s + (2 + 2i)7-s − 2.82i·11-s + (3 − 3i)13-s + (−2.82 + 2.82i)17-s + 8i·19-s + (2.82 + 2.82i)23-s + (−3.99 + 3i)25-s − 1.41·29-s + 4·31-s + (−2.82 + 5.65i)35-s + (3 + 3i)37-s − 9.89i·41-s + i·49-s + (2.82 + 2.82i)53-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 + (−0.685 + 0.685i)17-s + 1.83i·19-s + (0.589 + 0.589i)23-s + (−0.799 + 0.600i)25-s − 0.262·29-s + 0.718·31-s + (−0.478 + 0.956i)35-s + (0.493 + 0.493i)37-s − 1.54i·41-s + 0.142i·49-s + (0.388 + 0.388i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0799 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0799 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.093079020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.093079020\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.82 - 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.89iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (-2.82 - 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + (-4 - 4i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (-1 + i)T - 73iT^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 + (8.48 + 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.89T + 89T^{2} \) |
| 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
−8.661536749068096286910943174472, −8.345534424813178683297918581471, −7.54554994710587307903276968856, −6.53179152906679781952162174741, −5.75177833443346919381436300328, −5.50318288069425679045160222140, −4.02125785735336972221812949866, −3.32291430617390839319751358082, −2.34968966430422350903785234190, −1.35614319664647930855672401145,
0.70951641571912018364796833547, 1.68628533048522070505094576715, 2.70773815557248207783827620141, 4.33926174148688579169858580310, 4.48130764422735073818076733285, 5.26535313381007075782316190169, 6.50563007531549937738936554100, 7.00081610629350341639441307990, 7.908478895259045501873350073553, 8.693261999212309659651032932792
