| L(s) = 1 | − 1.41i·3-s + (1 + 2i)5-s + 4.82i·7-s + 0.999·9-s − 4.82·11-s + 0.585i·13-s + (2.82 − 1.41i)15-s − 2.82i·17-s − 19-s + 6.82·21-s + 7.65i·23-s + (−3 + 4i)25-s − 5.65i·27-s − 3.65·29-s − 6.82·31-s + ⋯ |
| L(s) = 1 | − 0.816i·3-s + (0.447 + 0.894i)5-s + 1.82i·7-s + 0.333·9-s − 1.45·11-s + 0.162i·13-s + (0.730 − 0.365i)15-s − 0.685i·17-s − 0.229·19-s + 1.49·21-s + 1.59i·23-s + (−0.600 + 0.800i)25-s − 1.08i·27-s − 0.679·29-s − 1.22·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.141039838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.141039838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 7 | \( 1 - 4.82iT - 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 0.585iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 23 | \( 1 - 7.65iT - 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 0.585iT - 37T^{2} \) |
| 41 | \( 1 - 0.828T + 41T^{2} \) |
| 43 | \( 1 + 8.82iT - 43T^{2} \) |
| 47 | \( 1 + 0.828iT - 47T^{2} \) |
| 53 | \( 1 - 11.8iT - 53T^{2} \) |
| 59 | \( 1 + 6.82T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 9.89iT - 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 - 1.17iT - 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 15.6iT - 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 1.07iT - 97T^{2} \) |
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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
−9.532884996198849169482186848452, −9.083396488697742850879467252025, −7.82484419048899405567620960496, −7.45563097884097312585651504506, −6.45414925598968588460083968723, −5.65481873053129725091626738544, −5.15389598807321655221649566183, −3.40030395237820032372923260713, −2.41739590564268304983929621571, −1.91137064637772366362919879439,
0.41770907532909417106655681277, 1.80805498302339497783668564638, 3.37469346335069180942106210771, 4.36236879414374898923380391109, 4.73297436810243206945379095644, 5.73559224303375853036450137298, 6.85584221348571251407790065550, 7.73581343124104883344593550113, 8.362189471259408177308344207814, 9.444842481773678421097485439773
