L(s) = 1 | − 1.41·2-s + (1.57 + 0.724i)3-s + 2.00·4-s + (−2.22 − 1.02i)6-s − 2.82·8-s + (1.94 + 2.28i)9-s − 6.61i·11-s + (3.14 + 1.44i)12-s + 4.00·16-s + 2.36·17-s + (−2.75 − 3.22i)18-s + 8.34·19-s + 9.34i·22-s + (−4.44 − 2.04i)24-s + (1.41 + 5.00i)27-s + ⋯ |
L(s) = 1 | − 1.00·2-s + (0.908 + 0.418i)3-s + 1.00·4-s + (−0.908 − 0.418i)6-s − 1.00·8-s + (0.649 + 0.760i)9-s − 1.99i·11-s + (0.908 + 0.418i)12-s + 1.00·16-s + 0.574·17-s + (−0.649 − 0.760i)18-s + 1.91·19-s + 1.99i·22-s + (−0.908 − 0.418i)24-s + (0.272 + 0.962i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37891 + 0.0220161i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37891 + 0.0220161i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (-1.57 - 0.724i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 6.61iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2.36T + 17T^{2} \) |
| 19 | \( 1 - 8.34T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 0.460iT - 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 14.3iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 13.6iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 12.7iT - 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−10.51677259953836415355806465482, −9.562551186872367704951546649217, −9.023905612785919070404255933228, −8.069659140466536902775725575019, −7.62282361436235098462207849687, −6.28879220793139465439898652868, −5.27395372844141817892382173442, −3.47027351198616670300548116129, −2.91359136674415451820286099810, −1.18312129097478287061811640227,
1.38371864787430730591459735101, 2.45068201346103903184227581432, 3.64004634526643748522909168060, 5.20109613221457762528898928516, 6.69238391431525245681093880565, 7.37607152908239710889903407975, 7.897233373907790234099118351323, 9.016472463233398359044224117172, 9.763246753311135177674704491143, 10.12677804972272662617560093710