L(s) = 1 | − 1.73i·3-s + i·5-s − 7-s + i·11-s + i·13-s + 1.73·15-s − 5.73·17-s + 2i·19-s + 1.73i·21-s + 3.46·23-s − 25-s − 5.19i·27-s + 9.19i·29-s − 4·31-s + 1.73·33-s + ⋯ |
L(s) = 1 | − 0.999i·3-s + 0.447i·5-s − 0.377·7-s + 0.301i·11-s + 0.277i·13-s + 0.447·15-s − 1.39·17-s + 0.458i·19-s + 0.377i·21-s + 0.722·23-s − 0.200·25-s − 1.00i·27-s + 1.70i·29-s − 0.718·31-s + 0.301·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9900435708\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9900435708\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 1.73iT - 3T^{2} \) |
| 11 | \( 1 - iT - 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 9.19iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 7.46iT - 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 3.92T + 47T^{2} \) |
| 53 | \( 1 - 5.46iT - 53T^{2} \) |
| 59 | \( 1 - 2.53iT - 59T^{2} \) |
| 61 | \( 1 - 5.46iT - 61T^{2} \) |
| 67 | \( 1 - 1.46iT - 67T^{2} \) |
| 71 | \( 1 + 0.535T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 5.73T + 79T^{2} \) |
| 83 | \( 1 - 15.4iT - 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−9.041147674251489566635383700842, −8.428466180177196288865104290441, −7.35837279754403128826554474912, −6.89647947769961117982641215379, −6.43709063223211639828688211839, −5.33092702040372285351025578045, −4.33490315593431359512513950388, −3.28573414485977212504911094074, −2.25948999928854333983943815111, −1.35279684342035068734097316451,
0.34036710664246468053305279547, 2.02905711773789271051580642974, 3.23023055982139221695062235033, 4.09401690912087438375180018375, 4.75301810028426929995802062698, 5.54597942805877710702293714472, 6.50186806838905702861415484266, 7.29984938292959839359321421257, 8.339522314348879261438668940665, 9.054938048483403578043351575829