| L(s) = 1 | + 1.66i·2-s − 3.28i·3-s − 0.757·4-s + 5.45·6-s + 0.418i·7-s + 2.06i·8-s − 7.79·9-s − 1.03·11-s + 2.48i·12-s − 1.83i·13-s − 0.694·14-s − 4.94·16-s − 5.10i·17-s − 12.9i·18-s − 3.10·19-s + ⋯ |
| L(s) = 1 | + 1.17i·2-s − 1.89i·3-s − 0.378·4-s + 2.22·6-s + 0.158i·7-s + 0.729i·8-s − 2.59·9-s − 0.312·11-s + 0.718i·12-s − 0.507i·13-s − 0.185·14-s − 1.23·16-s − 1.23i·17-s − 3.05i·18-s − 0.713·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8984455100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8984455100\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + iT \) |
| good | 2 | \( 1 - 1.66iT - 2T^{2} \) |
| 3 | \( 1 + 3.28iT - 3T^{2} \) |
| 7 | \( 1 - 0.418iT - 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 + 1.83iT - 13T^{2} \) |
| 17 | \( 1 + 5.10iT - 17T^{2} \) |
| 19 | \( 1 + 3.10T + 19T^{2} \) |
| 23 | \( 1 + 8.80iT - 23T^{2} \) |
| 29 | \( 1 - 1.66T + 29T^{2} \) |
| 31 | \( 1 + 3.86T + 31T^{2} \) |
| 37 | \( 1 + 5.86iT - 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 47 | \( 1 - 0.275iT - 47T^{2} \) |
| 53 | \( 1 - 7.35iT - 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 3.84T + 61T^{2} \) |
| 67 | \( 1 + 6.49iT - 67T^{2} \) |
| 71 | \( 1 - 6.25T + 71T^{2} \) |
| 73 | \( 1 - 2.78iT - 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 6.13iT - 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 9.11iT - 97T^{2} \) |
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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
−9.000929423223304309217146041921, −8.421278905532339068909250471826, −7.71190783533328549372001094623, −7.08080874629828851705226588549, −6.47052467011147329237069162778, −5.75675595445716720504748102917, −4.89532481591809702528781346023, −2.84621385636288221157365495897, −2.07418757425376902499733861889, −0.36665830568875921888044635002,
1.88572752180290116503982613461, 3.22380108843408285603012543401, 3.76463935106960658065998350052, 4.56709395166572214418170966039, 5.53369420143245135232342312855, 6.59804898761277708478454824224, 8.070666094537471052968979584163, 8.995221110557422598163137642684, 9.618644305229329096205343507064, 10.34166656771085283148628987936
