| L(s) = 1 | + 3.07i·3-s + (1.47 − 1.47i)7-s − 6.45·9-s + (1.20 − 1.20i)11-s + 5.63·13-s + (4.22 − 4.22i)17-s + (3.11 − 3.11i)19-s + (4.54 + 4.54i)21-s + (−1.08 − 1.08i)23-s − 10.6i·27-s + (5.32 + 5.32i)29-s + 4.67i·31-s + (3.71 + 3.71i)33-s − 1.51·37-s + 17.3i·39-s + ⋯ |
| L(s) = 1 | + 1.77i·3-s + (0.558 − 0.558i)7-s − 2.15·9-s + (0.363 − 0.363i)11-s + 1.56·13-s + (1.02 − 1.02i)17-s + (0.715 − 0.715i)19-s + (0.991 + 0.991i)21-s + (−0.225 − 0.225i)23-s − 2.04i·27-s + (0.988 + 0.988i)29-s + 0.839i·31-s + (0.646 + 0.646i)33-s − 0.248·37-s + 2.77i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.021952447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.021952447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3.07iT - 3T^{2} \) |
| 7 | \( 1 + (-1.47 + 1.47i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.20 + 1.20i)T - 11iT^{2} \) |
| 13 | \( 1 - 5.63T + 13T^{2} \) |
| 17 | \( 1 + (-4.22 + 4.22i)T - 17iT^{2} \) |
| 19 | \( 1 + (-3.11 + 3.11i)T - 19iT^{2} \) |
| 23 | \( 1 + (1.08 + 1.08i)T + 23iT^{2} \) |
| 29 | \( 1 + (-5.32 - 5.32i)T + 29iT^{2} \) |
| 31 | \( 1 - 4.67iT - 31T^{2} \) |
| 37 | \( 1 + 1.51T + 37T^{2} \) |
| 41 | \( 1 + 3.19iT - 41T^{2} \) |
| 43 | \( 1 - 2.42T + 43T^{2} \) |
| 47 | \( 1 + (-0.827 - 0.827i)T + 47iT^{2} \) |
| 53 | \( 1 + 8.17iT - 53T^{2} \) |
| 59 | \( 1 + (7.78 + 7.78i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.03 - 3.03i)T - 61iT^{2} \) |
| 67 | \( 1 - 2.93T + 67T^{2} \) |
| 71 | \( 1 - 0.180T + 71T^{2} \) |
| 73 | \( 1 + (-2.19 + 2.19i)T - 73iT^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 8.33iT - 83T^{2} \) |
| 89 | \( 1 + 9.08T + 89T^{2} \) |
| 97 | \( 1 + (6.04 - 6.04i)T - 97iT^{2} \) |
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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
−9.586733148125763802099793485408, −8.862978508324082622836490965904, −8.314887239290347138019953754519, −7.18604411604858779666258170744, −6.06235259842483473580880433719, −5.17999139629722437304880867828, −4.58874026268157152715137283407, −3.58942537258836019800766726899, −3.09005755270661970525463494598, −1.05483388727894644227150972705,
1.16227930618641208371901088680, 1.73968761353299037980488504975, 2.95075352309225645402900226053, 4.09820806684341025644856644502, 5.75628876187478651505872548587, 5.90806156345785792331689363418, 6.85279812877939059020668599119, 7.911755832221090323023695033962, 8.102974531040033145372030448904, 8.933598162800687316953464645043
