L(s) = 1 | + (−1.26 − 1.26i)3-s + (2.95 − 2.95i)5-s − i·7-s + 0.189i·9-s + (−3.18 + 3.18i)11-s + (3.42 + 3.42i)13-s − 7.46·15-s − 5.13·17-s + (−1.50 − 1.50i)19-s + (−1.26 + 1.26i)21-s − 7.11i·23-s − 12.4i·25-s + (−3.54 + 3.54i)27-s + (−3.84 − 3.84i)29-s − 0.831·31-s + ⋯ |
L(s) = 1 | + (−0.729 − 0.729i)3-s + (1.32 − 1.32i)5-s − 0.377i·7-s + 0.0630i·9-s + (−0.960 + 0.960i)11-s + (0.950 + 0.950i)13-s − 1.92·15-s − 1.24·17-s + (−0.345 − 0.345i)19-s + (−0.275 + 0.275i)21-s − 1.48i·23-s − 2.49i·25-s + (−0.683 + 0.683i)27-s + (−0.714 − 0.714i)29-s − 0.149·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9492141067\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9492141067\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (1.26 + 1.26i)T + 3iT^{2} \) |
| 5 | \( 1 + (-2.95 + 2.95i)T - 5iT^{2} \) |
| 11 | \( 1 + (3.18 - 3.18i)T - 11iT^{2} \) |
| 13 | \( 1 + (-3.42 - 3.42i)T + 13iT^{2} \) |
| 17 | \( 1 + 5.13T + 17T^{2} \) |
| 19 | \( 1 + (1.50 + 1.50i)T + 19iT^{2} \) |
| 23 | \( 1 + 7.11iT - 23T^{2} \) |
| 29 | \( 1 + (3.84 + 3.84i)T + 29iT^{2} \) |
| 31 | \( 1 + 0.831T + 31T^{2} \) |
| 37 | \( 1 + (5.64 - 5.64i)T - 37iT^{2} \) |
| 41 | \( 1 + 2.22iT - 41T^{2} \) |
| 43 | \( 1 + (-1.61 + 1.61i)T - 43iT^{2} \) |
| 47 | \( 1 + 7.83T + 47T^{2} \) |
| 53 | \( 1 + (-5.58 + 5.58i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.85 + 1.85i)T - 59iT^{2} \) |
| 61 | \( 1 + (1.65 + 1.65i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5.77 - 5.77i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.04iT - 71T^{2} \) |
| 73 | \( 1 + 7.67iT - 73T^{2} \) |
| 79 | \( 1 - 1.90T + 79T^{2} \) |
| 83 | \( 1 + (7.97 + 7.97i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.49iT - 89T^{2} \) |
| 97 | \( 1 + 1.98T + 97T^{2} \) |
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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
−8.859175542141613245195905702134, −8.260823299208740008071206593968, −6.89882863359796511858339903156, −6.53855037687944615098382799370, −5.68353480654021425653084989048, −4.86652544442032103554970637075, −4.21294301441118705573790910511, −2.23425971046131686668181412492, −1.63087195578963084039156341582, −0.35856906070082430401483563515,
1.87474941768895678449254964207, 2.89004474038949886633381455572, 3.69982391958596636661183466607, 5.24021197766554445959610962788, 5.66400947368433494386174953884, 6.15384921850716186816075911977, 7.14877180466442683111009466075, 8.168771419036755664778531464977, 9.112575799917745666816625553160, 9.900013112416010970291823948348