| L(s) = 1 | − 1.23i·3-s + i·7-s + 1.47·9-s − 4.23·11-s − 3.23i·13-s + 6.47i·17-s + 4.47·19-s + 1.23·21-s − 1.76i·23-s − 5.52i·27-s − 5·29-s + 9.70·31-s + 5.23i·33-s − 3i·37-s − 4.00·39-s + ⋯ |
| L(s) = 1 | − 0.713i·3-s + 0.377i·7-s + 0.490·9-s − 1.27·11-s − 0.897i·13-s + 1.56i·17-s + 1.02·19-s + 0.269·21-s − 0.367i·23-s − 1.06i·27-s − 0.928·29-s + 1.74·31-s + 0.911i·33-s − 0.493i·37-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.738712220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.738712220\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.23iT - 3T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 + 3.23iT - 13T^{2} \) |
| 17 | \( 1 - 6.47iT - 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 + 1.76iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 9.70T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 9.23T + 41T^{2} \) |
| 43 | \( 1 + 6.23iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 0.472iT - 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 - 0.236iT - 67T^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 + 13.2iT - 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 5.70iT - 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 0.763iT - 97T^{2} \) |
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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
−8.386284199246434097743642846071, −7.87002922988216978364986669936, −7.35928910229560201383264890094, −6.30577325107342222595726448687, −5.70883862085279828867330562576, −4.89403423048427344367072439043, −3.81232072359085965644846244769, −2.77160492027785213635822520288, −1.92631480164631027206484601485, −0.68986670435369074269108093697,
1.00440390636227903252348483389, 2.46606666472666573872426815570, 3.30912315600369660256099493713, 4.38524926996349585037815827913, 4.87552626524980226773160730441, 5.66521753198278851820034462698, 6.82939640554120448412844228575, 7.43044795999785003944791409801, 8.072527740393733618133650539977, 9.230359156755698774394162636328
