L(s) = 1 | + (2.18 − 2.18i)3-s + (−0.456 + 2.18i)5-s + (1.79 − 1.79i)7-s − 6.58i·9-s − 0.913·11-s + (−1.73 − 1.73i)13-s + (3.79 + 5.79i)15-s + (3 + 3i)17-s + 3.46i·19-s − 7.84i·21-s + (3.79 + 3.79i)23-s + (−4.58 − 1.99i)25-s + (−7.84 − 7.84i)27-s − 5.29·29-s − 7.58i·31-s + ⋯ |
L(s) = 1 | + (1.26 − 1.26i)3-s + (−0.204 + 0.978i)5-s + (0.677 − 0.677i)7-s − 2.19i·9-s − 0.275·11-s + (−0.480 − 0.480i)13-s + (0.978 + 1.49i)15-s + (0.727 + 0.727i)17-s + 0.794i·19-s − 1.71i·21-s + (0.790 + 0.790i)23-s + (−0.916 − 0.399i)25-s + (−1.50 − 1.50i)27-s − 0.982·29-s − 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.70330 - 0.888035i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.70330 - 0.888035i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.456 - 2.18i)T \) |
good | 3 | \( 1 + (-2.18 + 2.18i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1.79 + 1.79i)T - 7iT^{2} \) |
| 11 | \( 1 + 0.913T + 11T^{2} \) |
| 13 | \( 1 + (1.73 + 1.73i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3 - 3i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + (-3.79 - 3.79i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 + 7.58iT - 31T^{2} \) |
| 37 | \( 1 + (5.19 - 5.19i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.58T + 41T^{2} \) |
| 43 | \( 1 + (-0.361 + 0.361i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.79 - 3.79i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.64 - 2.64i)T + 53iT^{2} \) |
| 59 | \( 1 - 5.29iT - 59T^{2} \) |
| 61 | \( 1 - 6.20iT - 61T^{2} \) |
| 67 | \( 1 + (0.361 + 0.361i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.41iT - 71T^{2} \) |
| 73 | \( 1 + (8.58 - 8.58i)T - 73iT^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + (-3.10 + 3.10i)T - 83iT^{2} \) |
| 89 | \( 1 - 15.1iT - 89T^{2} \) |
| 97 | \( 1 + (8.58 + 8.58i)T + 97iT^{2} \) |
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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
−11.60673539132555124111742365151, −10.55159755226388338682878342731, −9.568337619037770038401931490101, −8.202299008647053676571322273917, −7.66980666709433218748833726674, −7.11370883195068802927833057079, −5.84367522706782448088251062787, −3.85544204918237329699425067864, −2.87327222323351124839965407479, −1.56856732259898553235872835770,
2.20688305333237280610987585714, 3.50072910890815620855250866064, 4.84907233032674634405910677312, 5.13438410011949035827611955586, 7.36951095107232986622605229937, 8.395930661539507612973823591481, 8.967099869425756385047019180556, 9.581870012834576800691672128071, 10.68969510583685538307049625836, 11.71340387028036167200249467648