| L(s) = 1 | − i·3-s + i·5-s − 3.60·7-s − 9-s − 5.78·11-s + 0.896·13-s + 15-s + 6.83i·17-s − 1.90·19-s + 3.60i·21-s + (−4.77 + 0.469i)23-s − 25-s + i·27-s + 1.39·29-s − 1.12i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447i·5-s − 1.36·7-s − 0.333·9-s − 1.74·11-s + 0.248·13-s + 0.258·15-s + 1.65i·17-s − 0.437·19-s + 0.786i·21-s + (−0.995 + 0.0979i)23-s − 0.200·25-s + 0.192i·27-s + 0.258·29-s − 0.201i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6626968489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6626968489\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (4.77 - 0.469i)T \) |
| good | 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + 5.78T + 11T^{2} \) |
| 13 | \( 1 - 0.896T + 13T^{2} \) |
| 17 | \( 1 - 6.83iT - 17T^{2} \) |
| 19 | \( 1 + 1.90T + 19T^{2} \) |
| 29 | \( 1 - 1.39T + 29T^{2} \) |
| 31 | \( 1 + 1.12iT - 31T^{2} \) |
| 37 | \( 1 + 7.48iT - 37T^{2} \) |
| 41 | \( 1 - 3.19T + 41T^{2} \) |
| 43 | \( 1 + 2.53T + 43T^{2} \) |
| 47 | \( 1 - 5.80iT - 47T^{2} \) |
| 53 | \( 1 - 2.67iT - 53T^{2} \) |
| 59 | \( 1 + 6.58iT - 59T^{2} \) |
| 61 | \( 1 - 0.597iT - 61T^{2} \) |
| 67 | \( 1 - 5.41T + 67T^{2} \) |
| 71 | \( 1 - 3.72iT - 71T^{2} \) |
| 73 | \( 1 - 0.479T + 73T^{2} \) |
| 79 | \( 1 + 1.03T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 5.21iT - 89T^{2} \) |
| 97 | \( 1 + 4.73iT - 97T^{2} \) |
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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
−7.972954928495248553248052790295, −7.39036290700180269373250260707, −6.47556322082707486572976773143, −6.06789494075943994654642794154, −5.44832668718976078253672642411, −4.19524241342502772527293531419, −3.42787144094339186962686920152, −2.64746735311724465905868517608, −1.91165111560392373581281161747, −0.29998447189454702584117069155,
0.55228490610993290831083658932, 2.33732007290723801108863315175, 2.96299917635788365259250278220, 3.73555242475831246822148668252, 4.76530223812250187969362157436, 5.24042066358006703381770813521, 6.05969394369134483800972648171, 6.76982065683138350456348861859, 7.64162895879431656338349762951, 8.294951927789323716306793191238
