| L(s) = 1 | + (1.73 + 0.0302i)3-s + 0.871·5-s + 3.09i·7-s + (2.99 + 0.104i)9-s + 1.90i·11-s + 1.23i·13-s + (1.50 + 0.0263i)15-s − 1.98i·17-s + 4.96·19-s + (−0.0933 + 5.35i)21-s + 23-s − 4.23·25-s + (5.18 + 0.271i)27-s − 2.31·29-s + 3.48i·31-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0174i)3-s + 0.389·5-s + 1.16i·7-s + (0.999 + 0.0348i)9-s + 0.573i·11-s + 0.343i·13-s + (0.389 + 0.00679i)15-s − 0.480i·17-s + 1.13·19-s + (−0.0203 + 1.16i)21-s + 0.208·23-s − 0.847·25-s + (0.998 + 0.0522i)27-s − 0.429·29-s + 0.625i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.839512533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.839512533\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.73 - 0.0302i)T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 0.871T + 5T^{2} \) |
| 7 | \( 1 - 3.09iT - 7T^{2} \) |
| 11 | \( 1 - 1.90iT - 11T^{2} \) |
| 13 | \( 1 - 1.23iT - 13T^{2} \) |
| 17 | \( 1 + 1.98iT - 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 29 | \( 1 + 2.31T + 29T^{2} \) |
| 31 | \( 1 - 3.48iT - 31T^{2} \) |
| 37 | \( 1 - 4.56iT - 37T^{2} \) |
| 41 | \( 1 + 8.71iT - 41T^{2} \) |
| 43 | \( 1 - 3.26T + 43T^{2} \) |
| 47 | \( 1 + 7.68T + 47T^{2} \) |
| 53 | \( 1 - 7.53T + 53T^{2} \) |
| 59 | \( 1 - 12.2iT - 59T^{2} \) |
| 61 | \( 1 - 10.7iT - 61T^{2} \) |
| 67 | \( 1 - 0.0562T + 67T^{2} \) |
| 71 | \( 1 + 7.84T + 71T^{2} \) |
| 73 | \( 1 + 7.04T + 73T^{2} \) |
| 79 | \( 1 - 5.10iT - 79T^{2} \) |
| 83 | \( 1 - 0.795iT - 83T^{2} \) |
| 89 | \( 1 + 17.3iT - 89T^{2} \) |
| 97 | \( 1 - 9.05T + 97T^{2} \) |
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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
−8.966244370334071660812568075373, −8.746753539168447119493150823682, −7.55452417834798732959182586215, −7.11653752704858707357158077220, −5.95774031550514809569733582909, −5.21924662611711561435076542919, −4.26916024812433023008290166928, −3.16977228032794386908346570606, −2.40984707447075556807161771108, −1.54165088855169069397428714549,
0.910304375206241497734866801899, 2.01781156558248032859605449416, 3.23772856761617943211200888096, 3.78984719025595877047631640806, 4.77015901597561111148419461791, 5.84012282903547991474079466903, 6.73175449937227201148158842531, 7.66872174628804174946811955340, 7.935400466034620003505620363123, 8.987905257125049352542678196258
