L(s) = 1 | + (1.41 − 1.73i)5-s + 1.41i·7-s − 2i·11-s + 5.65·13-s + 4.89i·17-s + 6i·19-s + 7.07i·23-s + (−0.999 − 4.89i)25-s + 6.92i·29-s − 6.92·31-s + (2.44 + 2.00i)35-s − 2.82·37-s + 4·41-s + 2.44·43-s − 4.24i·47-s + ⋯ |
L(s) = 1 | + (0.632 − 0.774i)5-s + 0.534i·7-s − 0.603i·11-s + 1.56·13-s + 1.18i·17-s + 1.37i·19-s + 1.47i·23-s + (−0.199 − 0.979i)25-s + 1.28i·29-s − 1.24·31-s + (0.414 + 0.338i)35-s − 0.464·37-s + 0.624·41-s + 0.373·43-s − 0.618i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.123442781\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.123442781\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.41 + 1.73i)T \) |
good | 7 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 7.07iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 + 4.24iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 2iT - 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 2.44T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 14.6iT - 97T^{2} \) |
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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
−8.809721510860158043338473479094, −8.346630474119744999305693824271, −7.46890314570757043890228757561, −6.18193455264211316181956903206, −5.83836947724165363140042949661, −5.25780617675567444320544055955, −3.91281465527723179056666926051, −3.42580489207215935340862246928, −1.87265941010813181869833285611, −1.27425409194433893991340328915,
0.73166397763057965766537559657, 2.10781748291682981416453662346, 2.90889976952843919419602505395, 3.93250541707485127845257778272, 4.75695656151853090780347608050, 5.75971600106519686296844382730, 6.52944415646888160417704332877, 7.06357674679749392055555482727, 7.80240132205881316978325244789, 8.895387277324523679724873080418