L(s) = 1 | + 0.414i·2-s + 3-s + 1.82·4-s − 2.82i·5-s + 0.414i·6-s − 2.82i·7-s + 1.58i·8-s + 9-s + 1.17·10-s + 2i·11-s + 1.82·12-s + 1.17·14-s − 2.82i·15-s + 3·16-s − 7.65·17-s + 0.414i·18-s + ⋯ |
L(s) = 1 | + 0.292i·2-s + 0.577·3-s + 0.914·4-s − 1.26i·5-s + 0.169i·6-s − 1.06i·7-s + 0.560i·8-s + 0.333·9-s + 0.370·10-s + 0.603i·11-s + 0.527·12-s + 0.313·14-s − 0.730i·15-s + 0.750·16-s − 1.85·17-s + 0.0976i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.98701 - 0.601620i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.98701 - 0.601620i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.414iT - 2T^{2} \) |
| 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 1.17iT - 31T^{2} \) |
| 37 | \( 1 - 7.65iT - 37T^{2} \) |
| 41 | \( 1 - 5.17iT - 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 11.6iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 7.65iT - 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 6.82iT - 67T^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 + 0.343iT - 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 3.65iT - 83T^{2} \) |
| 89 | \( 1 + 14.8iT - 89T^{2} \) |
| 97 | \( 1 - 3.65iT - 97T^{2} \) |
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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
−10.88647147366311296319805408955, −9.844237409722385703182190240172, −8.902989161135973426812215123721, −8.130782279648337196236388229706, −7.16113606393434282866593448667, −6.52255265429510025199285516384, −4.94758832672999984264911150930, −4.26097734204410416675861396163, −2.65700944160400447192490663477, −1.29298295527238399051943004688,
2.15222466683602588959249188618, 2.75531128747965607338796488580, 3.79239884410095802067850699850, 5.60603661368182320709479726436, 6.60311533599608811756224199124, 7.15576024923931373735185368729, 8.400504597629620139897427514785, 9.182374177958968420282998651461, 10.38928897116044569767685879308, 10.95145106171776430141403217366