L(s) = 1 | + 3-s − 5-s + 3.46i·7-s + 9-s + 3.46i·11-s − 15-s + 6·17-s + (−4 + 1.73i)19-s + 3.46i·21-s + 3.46i·23-s + 25-s + 27-s − 8·31-s + 3.46i·33-s − 3.46i·35-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.30i·7-s + 0.333·9-s + 1.04i·11-s − 0.258·15-s + 1.45·17-s + (−0.917 + 0.397i)19-s + 0.755i·21-s + 0.722i·23-s + 0.200·25-s + 0.192·27-s − 1.43·31-s + 0.603i·33-s − 0.585i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.505931395\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.505931395\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (4 - 1.73i)T \) |
good | 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 10.3iT - 47T^{2} \) |
| 53 | \( 1 + 6.92iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−8.460152976981706593284473992816, −8.018598223541456384868013921902, −7.32103868569831823304225124888, −6.49953479370163999640792460265, −5.57127623534603184219590009146, −5.00174978450755419537210465904, −3.95722762751561373908472329755, −3.26603766495715402476871054369, −2.30682750869853301897391270928, −1.55613390707426015664982458515,
0.38293732144677557968102448870, 1.37372377307192680778531789547, 2.71150527587835831433144110884, 3.58178377958223990297484956940, 4.01008805816627195268461206719, 4.93889407390658293860696851379, 5.92319434153301586199589305131, 6.72602237547330614174529718153, 7.56647856111963016942641309268, 7.85094560334792266325663344704