| L(s) = 1 | − 5-s + 5.20i·11-s + 2.02i·13-s − 4.07·17-s + 0.815i·19-s − 7.37i·23-s + 25-s − 1.50i·29-s − 10.0i·31-s + 4.00·37-s + 3.30·41-s + 7.47·43-s + 6.61·47-s − 8.97i·53-s − 5.20i·55-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.57i·11-s + 0.561i·13-s − 0.988·17-s + 0.187i·19-s − 1.53i·23-s + 0.200·25-s − 0.279i·29-s − 1.81i·31-s + 0.658·37-s + 0.515·41-s + 1.14·43-s + 0.965·47-s − 1.23i·53-s − 0.702i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.526056094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.526056094\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 5.20iT - 11T^{2} \) |
| 13 | \( 1 - 2.02iT - 13T^{2} \) |
| 17 | \( 1 + 4.07T + 17T^{2} \) |
| 19 | \( 1 - 0.815iT - 19T^{2} \) |
| 23 | \( 1 + 7.37iT - 23T^{2} \) |
| 29 | \( 1 + 1.50iT - 29T^{2} \) |
| 31 | \( 1 + 10.0iT - 31T^{2} \) |
| 37 | \( 1 - 4.00T + 37T^{2} \) |
| 41 | \( 1 - 3.30T + 41T^{2} \) |
| 43 | \( 1 - 7.47T + 43T^{2} \) |
| 47 | \( 1 - 6.61T + 47T^{2} \) |
| 53 | \( 1 + 8.97iT - 53T^{2} \) |
| 59 | \( 1 + 7.67T + 59T^{2} \) |
| 61 | \( 1 + 10.0iT - 61T^{2} \) |
| 67 | \( 1 - 2.05T + 67T^{2} \) |
| 71 | \( 1 - 6.47iT - 71T^{2} \) |
| 73 | \( 1 - 11.9iT - 73T^{2} \) |
| 79 | \( 1 - 5.61T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 2.14T + 89T^{2} \) |
| 97 | \( 1 - 17.1iT - 97T^{2} \) |
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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
−7.76838013065595392938329395863, −7.17042260612697632231550347105, −6.55624069876323211465959341630, −5.90734605255920463440483682884, −4.75911077598701583436691386170, −4.39669061581543101459894242464, −3.81823058688403560326799477663, −2.39358463645446586580768982324, −2.17888566667302771174342665022, −0.70921426319946553571939709467,
0.51718382220880686396917511109, 1.47484320468127888511685169981, 2.80337584973289482074997661278, 3.26301280078834151925915486814, 4.10028435197096220863102803919, 4.88062141414055121584740263941, 5.76673436882356918067222731391, 6.11062130831097471124147862470, 7.17094968494260541774372546534, 7.57867846166903197714767767539
