L(s) = 1 | + (0.5 − 0.866i)2-s + (1 + 1.73i)3-s + (−0.499 − 0.866i)4-s + 1.99·6-s + (2.5 − 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.999 − 1.73i)12-s + 2·13-s + (0.500 − 2.59i)14-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s + (0.499 + 0.866i)18-s + (−4 + 6.92i)19-s + (4 + 3.46i)21-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.577 + 0.999i)3-s + (−0.249 − 0.433i)4-s + 0.816·6-s + (0.944 − 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.288 − 0.499i)12-s + 0.554·13-s + (0.133 − 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + (0.117 + 0.204i)18-s + (−0.917 + 1.58i)19-s + (0.872 + 0.755i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.00692 - 0.127359i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.00692 - 0.127359i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 3 | \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.5 + 7.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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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
−11.24687569770698452644439309712, −10.50786472149861832151907270950, −9.922335082150076051493525082542, −8.691442805745142106916194205125, −8.152060182224736353905691858016, −6.48628705948829659225934705940, −5.15427063611177268782455829864, −4.17571636462357330071331152354, −3.44214755510998225275310245632, −1.76564402247031024152574226589,
1.68486586956382681209348921118, 3.10207045712088492456262227600, 4.70392693746425574624794487791, 5.66777547172578367733454129336, 7.07265906853716985985947101983, 7.46244244242676405754593605610, 8.593873130595189722389972798567, 9.115901610106276632386415197831, 10.85490003450880797361211485432, 11.64144106565294631226270546173