| L(s) = 1 | + (−0.142 + 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (1.69 + 1.96i)5-s + (−0.959 + 0.281i)6-s + (−1.04 − 0.668i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (−2.18 + 1.40i)10-s + (0.0886 + 0.616i)11-s + (−0.142 − 0.989i)12-s + (−1.11 + 0.716i)13-s + (0.809 − 0.934i)14-s + (−1.07 + 2.35i)15-s + (0.841 + 0.540i)16-s + (4.51 − 1.32i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (0.759 + 0.876i)5-s + (−0.391 + 0.115i)6-s + (−0.393 − 0.252i)7-s + (0.146 − 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.690 + 0.443i)10-s + (0.0267 + 0.185i)11-s + (−0.0410 − 0.285i)12-s + (−0.309 + 0.198i)13-s + (0.216 − 0.249i)14-s + (−0.278 + 0.609i)15-s + (0.210 + 0.135i)16-s + (1.09 − 0.321i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.119 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.752173 + 0.848035i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.752173 + 0.848035i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-3.35 + 3.42i)T \) |
| good | 5 | \( 1 + (-1.69 - 1.96i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (1.04 + 0.668i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.0886 - 0.616i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.11 - 0.716i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.51 + 1.32i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (3.63 + 1.06i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.06 + 0.313i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.92 + 8.60i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.06 + 1.22i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-7.75 - 8.95i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.697 + 1.52i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 6.94T + 47T^{2} \) |
| 53 | \( 1 + (9.46 + 6.08i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (8.19 - 5.26i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (5.97 - 13.0i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.871 + 6.05i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.923 - 6.42i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-12.5 - 3.68i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (10.1 - 6.53i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (4.83 - 5.58i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-1.40 - 3.07i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (7.49 + 8.65i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−13.77169446714086757888648909338, −12.74227192448554662189399901063, −11.14643826725540527981095538963, −10.03839262177207696451723304080, −9.543561004507738076124003324135, −8.124858924693991691456885917045, −6.88561681741852977204941565898, −5.96421444950631544619855469793, −4.48902131847679200972959817240, −2.81424796643085917373179591481,
1.48692967494587366174879011370, 3.14761353504357043299672795388, 5.00606364192275380374107413463, 6.19802338769429819636985698651, 7.85554516662302576460998100211, 8.938286869925637464303633218734, 9.692519410598239571553661136616, 10.84943227733832896510331237819, 12.36784359200491011014379266998, 12.62519575596215070295793760734
