| L(s) = 1 | − 3.46i·5-s + 3.46i·7-s + 3·11-s + 4·13-s − 1.73i·17-s + 1.73i·19-s − 6.99·25-s + 3.46i·29-s + 11.9·35-s + 2·37-s − 5.19i·41-s − 5.19i·43-s + 12·47-s − 4.99·49-s − 10.3i·55-s + ⋯ |
| L(s) = 1 | − 1.54i·5-s + 1.30i·7-s + 0.904·11-s + 1.10·13-s − 0.420i·17-s + 0.397i·19-s − 1.39·25-s + 0.643i·29-s + 2.02·35-s + 0.328·37-s − 0.811i·41-s − 0.792i·43-s + 1.75·47-s − 0.714·49-s − 1.40i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.831114332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.831114332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 5.19iT - 41T^{2} \) |
| 43 | \( 1 + 5.19iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 15T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 8.66iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 13.8iT - 89T^{2} \) |
| 97 | \( 1 - 13T + 97T^{2} \) |
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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
−9.178466955321563983536739355857, −8.840227991322439352832031397699, −8.386830115888109776412920370753, −7.14930385665870715393193873159, −5.93213462606578208720240308081, −5.52551304385686249930763676027, −4.51108644376645875118903825586, −3.58340207176620914205424490886, −2.09386641525624104325426519838, −1.00161296328331945010126619260,
1.14548917146197857190991517174, 2.65489835237128257274058284684, 3.75233956916435402674675679681, 4.17208758990055764977770608121, 5.84105446650629553414329101778, 6.60443059315058874371763263857, 7.09263305290586059048594295861, 7.929805824135421689408514952505, 8.935080117476293169730794564247, 10.03434297233564260252939438253
