L(s) = 1 | + (1.46 + 0.391i)2-s + (−0.852 + 1.50i)3-s + (0.246 + 0.142i)4-s + (0.207 + 2.22i)5-s + (−1.83 + 1.86i)6-s + (2.36 − 1.17i)7-s + (−1.83 − 1.83i)8-s + (−1.54 − 2.57i)9-s + (−0.567 + 3.33i)10-s + (0.791 + 0.457i)11-s + (−0.425 + 0.250i)12-s + (3.07 − 3.07i)13-s + (3.92 − 0.791i)14-s + (−3.53 − 1.58i)15-s + (−2.24 − 3.88i)16-s + (−0.311 − 1.16i)17-s + ⋯ |
L(s) = 1 | + (1.03 + 0.276i)2-s + (−0.492 + 0.870i)3-s + (0.123 + 0.0712i)4-s + (0.0929 + 0.995i)5-s + (−0.749 + 0.762i)6-s + (0.895 − 0.444i)7-s + (−0.648 − 0.648i)8-s + (−0.514 − 0.857i)9-s + (−0.179 + 1.05i)10-s + (0.238 + 0.137i)11-s + (−0.122 + 0.0723i)12-s + (0.854 − 0.854i)13-s + (1.04 − 0.211i)14-s + (−0.912 − 0.409i)15-s + (−0.561 − 0.971i)16-s + (−0.0755 − 0.281i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.494 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20792 + 0.702101i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20792 + 0.702101i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.852 - 1.50i)T \) |
| 5 | \( 1 + (-0.207 - 2.22i)T \) |
| 7 | \( 1 + (-2.36 + 1.17i)T \) |
good | 2 | \( 1 + (-1.46 - 0.391i)T + (1.73 + i)T^{2} \) |
| 11 | \( 1 + (-0.791 - 0.457i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.07 + 3.07i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.311 + 1.16i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (5.95 - 3.43i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.505 + 1.88i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 2.72T + 29T^{2} \) |
| 31 | \( 1 + (2.31 - 4.01i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.207 + 0.774i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 0.922iT - 41T^{2} \) |
| 43 | \( 1 + (4.80 - 4.80i)T - 43iT^{2} \) |
| 47 | \( 1 + (10.1 + 2.71i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-10.6 + 2.85i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (4.94 - 8.55i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.533 - 0.924i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.83 + 1.83i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 0.557iT - 71T^{2} \) |
| 73 | \( 1 + (0.564 + 2.10i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (2.62 - 1.51i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.38 + 2.38i)T + 83iT^{2} \) |
| 89 | \( 1 + (5.64 + 9.78i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.58 + 1.58i)T + 97iT^{2} \) |
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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
−14.30272210670142425453303565138, −13.11110177819237793458065945660, −11.81189390027989181928474375312, −10.78815951217563693881898825013, −10.07988454190114937277996752450, −8.487376690451943609342417553245, −6.72204925022024930630197013023, −5.74923862391289874919269195721, −4.51432204155040616967203390489, −3.44771812668954614786566177576,
1.94251051800903210010206832643, 4.29748543976028554220766308479, 5.31037862863709781698238387966, 6.38816928945126939974966677221, 8.256074449985967076812318599343, 8.889221555283975603234675284921, 11.15114605802926037586254613799, 11.73712095609489215452410432687, 12.68660767633995106029380051960, 13.37685217783887738515015390753