L(s) = 1 | + 2.82·2-s − 5.19·3-s + 8.00·4-s − 19.9i·5-s − 14.6·6-s + 58.9i·7-s + 22.6·8-s + 27·9-s − 56.4i·10-s − 168. i·11-s − 41.5·12-s + 288.·13-s + 166. i·14-s + 103. i·15-s + 64.0·16-s − 311. i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.500·4-s − 0.798i·5-s − 0.408·6-s + 1.20i·7-s + 0.353·8-s + 0.333·9-s − 0.564i·10-s − 1.39i·11-s − 0.288·12-s + 1.70·13-s + 0.851i·14-s + 0.461i·15-s + 0.250·16-s − 1.07i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.23012 - 0.889732i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.23012 - 0.889732i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2.82T \) |
| 3 | \( 1 + 5.19T \) |
| 23 | \( 1 + (383. + 364. i)T \) |
good | 5 | \( 1 + 19.9iT - 625T^{2} \) |
| 7 | \( 1 - 58.9iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 168. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 288.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 311. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 118. iT - 1.30e5T^{2} \) |
| 29 | \( 1 - 991.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 827.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 80.7iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 187.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 331. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.89e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 4.19e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 4.96e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 5.56e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 6.83e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 1.91e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 6.60e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.13e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 3.57e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.64e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.50e3iT - 8.85e7T^{2} \) |
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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
−12.27293279607593286418713827260, −11.61543443674230444885347864985, −10.68942151760235742706757663930, −9.007682593493621209139396170933, −8.308054466014742429221851812419, −6.35853853434091387505497990989, −5.71539552784577248064941730206, −4.59021496278937717358215991100, −2.95974864191187110722597479501, −0.963779725728799674125977155099,
1.49763274861329333310038374561, 3.55781454011175491406763053560, 4.51870727710932584281063573038, 6.17404835844895887399680991207, 6.86272954025787954172798826223, 8.025540372380865120375579582756, 10.08526642983134438314090413411, 10.61517230199706229461997034734, 11.56257743524282506251009591122, 12.70749839934523854729228983836