L(s) = 1 | − i·2-s + 3i·3-s − 4-s + 3·6-s + i·8-s − 6·9-s − 3·11-s − 3i·12-s + i·13-s + 16-s + 7i·17-s + 6i·18-s − 19-s + 3i·22-s − 4i·23-s − 3·24-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.73i·3-s − 0.5·4-s + 1.22·6-s + 0.353i·8-s − 2·9-s − 0.904·11-s − 0.866i·12-s + 0.277i·13-s + 0.250·16-s + 1.69i·17-s + 1.41i·18-s − 0.229·19-s + 0.639i·22-s − 0.834i·23-s − 0.612·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.159754 + 0.676729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.159754 + 0.676729i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 3 | \( 1 - 3iT - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 17 | \( 1 - 7iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 12iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 15iT - 83T^{2} \) |
| 89 | \( 1 - 11T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−10.68289644424476656665108512165, −10.34305563222597825249071363127, −9.316346563541359854773596507880, −8.765309970270127063388160474161, −7.75129832658760489349632252111, −6.00848546992875967342402373826, −5.19715025946814237771748071807, −4.18935121509668318593566519653, −3.57324414348872353957985510826, −2.26825023674913546597657460255,
0.35568185536579578587627323658, 1.99206581273845457291176522796, 3.25780577683835846502962046170, 5.15326324408768038062307884309, 5.73412812447858352593367456974, 7.05880187414150937079861211622, 7.22373060948437650523184381463, 8.156013622075974525383855662152, 8.937054047937670167177999070291, 10.06936877878986749585616041677