L(s) = 1 | + (−0.173 + 0.300i)2-s + (−0.5 − 0.866i)3-s + (0.439 + 0.761i)4-s + (−0.766 − 1.32i)5-s + 0.347·6-s − 0.652·8-s + (−0.499 + 0.866i)9-s + 0.532·10-s + (−0.766 + 1.32i)11-s + (0.439 − 0.761i)12-s + (−0.766 + 1.32i)15-s + (−0.326 + 0.565i)16-s + 0.347·17-s + (−0.173 − 0.300i)18-s + (0.673 − 1.16i)20-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.300i)2-s + (−0.5 − 0.866i)3-s + (0.439 + 0.761i)4-s + (−0.766 − 1.32i)5-s + 0.347·6-s − 0.652·8-s + (−0.499 + 0.866i)9-s + 0.532·10-s + (−0.766 + 1.32i)11-s + (0.439 − 0.761i)12-s + (−0.766 + 1.32i)15-s + (−0.326 + 0.565i)16-s + 0.347·17-s + (−0.173 − 0.300i)18-s + (0.673 − 1.16i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5469326107\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5469326107\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 239 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 0.347T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 2T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.091635431971645287767364131539, −8.348647224336118241736583286181, −7.902433353261969168612495454980, −7.18570804164188579899646107501, −6.65205735327496845885230673698, −5.31891866304058869810316080529, −4.90239767203338450683981028336, −3.75577205103329397409624001980, −2.51953775668099392409197070350, −1.37011958547190538091790616067,
0.43921422556257897203297883536, 2.50437819724785868257449669641, 3.20314045939619209632830680398, 4.03140446766330096256140806525, 5.23455186336868272654657503017, 6.06132509634061456436367148611, 6.43531100875138214053300807566, 7.59792183450338762473195153687, 8.277763414384457435248483538306, 9.508798982786617044286688012398