| L(s) = 1 | − 2.23i·5-s − 3.87i·7-s − 1.73·11-s − 2·13-s − 4.47i·17-s − 6.92·23-s + 4.47i·29-s + 3.87i·31-s − 8.66·35-s + 4·37-s + 8.94i·41-s − 7.74i·43-s − 3.46·47-s − 8.00·49-s − 2.23i·53-s + ⋯ |
| L(s) = 1 | − 0.999i·5-s − 1.46i·7-s − 0.522·11-s − 0.554·13-s − 1.08i·17-s − 1.44·23-s + 0.830i·29-s + 0.695i·31-s − 1.46·35-s + 0.657·37-s + 1.39i·41-s − 1.18i·43-s − 0.505·47-s − 1.14·49-s − 0.307i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8379584553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8379584553\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.23iT - 5T^{2} \) |
| 7 | \( 1 + 3.87iT - 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 - 4.47iT - 29T^{2} \) |
| 31 | \( 1 - 3.87iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 8.94iT - 41T^{2} \) |
| 43 | \( 1 + 7.74iT - 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 2.23iT - 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 7.74iT - 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 - 7.74iT - 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 4.47iT - 89T^{2} \) |
| 97 | \( 1 - 11T + 97T^{2} \) |
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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
−8.903924501709111508943579648491, −8.074362320036344303505657708236, −7.40460137676977333962314835880, −6.69866474754708933470649130867, −5.45624154831014595108225179134, −4.74342490083514527970801202610, −4.06792729940957682415109316632, −2.89097057270423967486267410496, −1.44122967911840906743579244448, −0.30718620337990413194323516852,
2.08617075506423563662371953794, 2.63601843455134035974488438199, 3.73021094408857367259475447845, 4.87356676384399571926684490045, 6.01832541292989651788101354156, 6.17047392633193031559096400435, 7.46920201193895123316003397862, 8.077061751780683620669816171409, 8.910505547143542810093234296352, 9.806409758342984961893525753998
