L(s) = 1 | + (−0.244 − 1.39i)2-s + 2.91·3-s + (−1.88 + 0.682i)4-s + (1.83 + 1.27i)5-s + (−0.712 − 4.05i)6-s + (−1.52 + 2.16i)7-s + (1.41 + 2.45i)8-s + 5.47·9-s + (1.32 − 2.87i)10-s − 0.0929·11-s + (−5.47 + 1.98i)12-s − 4.08i·13-s + (3.38 + 1.59i)14-s + (5.34 + 3.71i)15-s + (3.06 − 2.56i)16-s − 4.24·17-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)2-s + 1.68·3-s + (−0.940 + 0.341i)4-s + (0.821 + 0.570i)5-s + (−0.290 − 1.65i)6-s + (−0.575 + 0.817i)7-s + (0.498 + 0.866i)8-s + 1.82·9-s + (0.419 − 0.907i)10-s − 0.0280·11-s + (−1.57 + 0.573i)12-s − 1.13i·13-s + (0.905 + 0.425i)14-s + (1.38 + 0.958i)15-s + (0.767 − 0.641i)16-s − 1.02·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.77116 - 0.644609i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.77116 - 0.644609i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.244 + 1.39i)T \) |
| 5 | \( 1 + (-1.83 - 1.27i)T \) |
| 7 | \( 1 + (1.52 - 2.16i)T \) |
good | 3 | \( 1 - 2.91T + 3T^{2} \) |
| 11 | \( 1 + 0.0929T + 11T^{2} \) |
| 13 | \( 1 + 4.08iT - 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 + 2.39iT - 19T^{2} \) |
| 23 | \( 1 + 4.52T + 23T^{2} \) |
| 29 | \( 1 + 4.35iT - 29T^{2} \) |
| 31 | \( 1 - 1.10T + 31T^{2} \) |
| 37 | \( 1 + 8.54T + 37T^{2} \) |
| 41 | \( 1 + 6.10iT - 41T^{2} \) |
| 43 | \( 1 - 4.60iT - 43T^{2} \) |
| 47 | \( 1 + 7.93iT - 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 - 10.6iT - 59T^{2} \) |
| 61 | \( 1 - 1.92T + 61T^{2} \) |
| 67 | \( 1 - 13.1iT - 67T^{2} \) |
| 71 | \( 1 - 9.75iT - 71T^{2} \) |
| 73 | \( 1 - 6.19T + 73T^{2} \) |
| 79 | \( 1 - 3.42iT - 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 4.46iT - 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−11.81604368675057860450391155669, −10.39368476157307831423040927671, −9.896156415530960325889165487571, −8.933533906214521584452147964497, −8.437889032125969330386561006548, −7.12653844248642308988467779438, −5.55992015755094388886302883552, −3.84491186230156538763447491072, −2.74834450722456813924512082117, −2.20210922555538196783142372772,
1.83293006306336516710110993277, 3.69589526367582647660260983399, 4.65707570259048015695216712143, 6.33088615289996760501387553822, 7.16033937473296063082836656884, 8.256840982294562046693977794161, 9.009053969241994765484687223471, 9.608366602580082178019532051441, 10.41042929355513784338869846561, 12.53121444797238722849217535568