| L(s) = 1 | + (1.66 + 1.66i)3-s − 2.89·7-s + 2.53i·9-s + (−1.84 − 1.84i)11-s + (−3.08 − 3.08i)13-s − 7.29i·17-s + (−1.23 + 1.23i)19-s + (−4.81 − 4.81i)21-s − 4.60·23-s + (0.772 − 0.772i)27-s + (−4.24 + 4.24i)29-s − 2.06·31-s − 6.13i·33-s + (−1.17 + 1.17i)37-s − 10.2i·39-s + ⋯ |
| L(s) = 1 | + (0.960 + 0.960i)3-s − 1.09·7-s + 0.845i·9-s + (−0.556 − 0.556i)11-s + (−0.854 − 0.854i)13-s − 1.77i·17-s + (−0.283 + 0.283i)19-s + (−1.05 − 1.05i)21-s − 0.960·23-s + (0.148 − 0.148i)27-s + (−0.788 + 0.788i)29-s − 0.370·31-s − 1.06i·33-s + (−0.193 + 0.193i)37-s − 1.64i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6599163033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6599163033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-1.66 - 1.66i)T + 3iT^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 11 | \( 1 + (1.84 + 1.84i)T + 11iT^{2} \) |
| 13 | \( 1 + (3.08 + 3.08i)T + 13iT^{2} \) |
| 17 | \( 1 + 7.29iT - 17T^{2} \) |
| 19 | \( 1 + (1.23 - 1.23i)T - 19iT^{2} \) |
| 23 | \( 1 + 4.60T + 23T^{2} \) |
| 29 | \( 1 + (4.24 - 4.24i)T - 29iT^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 + (1.17 - 1.17i)T - 37iT^{2} \) |
| 41 | \( 1 - 4.61iT - 41T^{2} \) |
| 43 | \( 1 + (-3.03 + 3.03i)T - 43iT^{2} \) |
| 47 | \( 1 + 11.7iT - 47T^{2} \) |
| 53 | \( 1 + (2.73 - 2.73i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3.11 - 3.11i)T + 59iT^{2} \) |
| 61 | \( 1 + (-2.34 + 2.34i)T - 61iT^{2} \) |
| 67 | \( 1 + (-8.24 - 8.24i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.25iT - 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 0.113T + 79T^{2} \) |
| 83 | \( 1 + (9.76 + 9.76i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.74iT - 89T^{2} \) |
| 97 | \( 1 - 13.9iT - 97T^{2} \) |
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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
−9.248860969448982515610731755110, −8.578201618761350338697202938956, −7.68334300594201710191720876691, −6.91610482850492997999585186348, −5.71420601853055690156369312544, −4.98447266136090402171567078152, −3.85294611625827139097319175053, −3.10495281676471902882149907734, −2.50550590178057378780580400006, −0.20121255918702067199269823180,
1.81422695431656213676358588227, 2.42723524664871557265862591316, 3.53847747860404609982844896897, 4.44629300562442268265621781516, 5.86319264288132594707654431020, 6.57169703499242515163738150388, 7.36481712570833262204509705941, 7.940450561856141933318074007926, 8.782992766902846540454424585525, 9.567695182244820318006947238244
