| L(s) = 1 | + (−0.883 + 1.93i)2-s + (0.959 + 0.281i)3-s + (−1.65 − 1.90i)4-s + (−0.841 + 0.540i)5-s + (−1.39 + 1.60i)6-s + (−0.350 + 2.44i)7-s + (1.07 − 0.314i)8-s + (0.841 + 0.540i)9-s + (−0.302 − 2.10i)10-s + (1.03 + 2.27i)11-s + (−1.04 − 2.29i)12-s + (−0.0750 − 0.521i)13-s + (−4.41 − 2.83i)14-s + (−0.959 + 0.281i)15-s + (0.380 − 2.64i)16-s + (−2.99 + 3.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.624 + 1.36i)2-s + (0.553 + 0.162i)3-s + (−0.826 − 0.954i)4-s + (−0.376 + 0.241i)5-s + (−0.568 + 0.656i)6-s + (−0.132 + 0.922i)7-s + (0.379 − 0.111i)8-s + (0.280 + 0.180i)9-s + (−0.0957 − 0.665i)10-s + (0.312 + 0.685i)11-s + (−0.302 − 0.663i)12-s + (−0.0208 − 0.144i)13-s + (−1.17 − 0.757i)14-s + (−0.247 + 0.0727i)15-s + (0.0950 − 0.661i)16-s + (−0.726 + 0.837i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.112487 - 0.831533i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112487 - 0.831533i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 5 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (2.17 - 4.27i)T \) |
| good | 2 | \( 1 + (0.883 - 1.93i)T + (-1.30 - 1.51i)T^{2} \) |
| 7 | \( 1 + (0.350 - 2.44i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.03 - 2.27i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.0750 + 0.521i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (2.99 - 3.45i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (3.29 + 3.80i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (0.853 - 0.985i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-1.00 + 0.296i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (3.98 + 2.55i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (2.07 - 1.33i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-11.7 - 3.44i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 6.76T + 47T^{2} \) |
| 53 | \( 1 + (-0.343 + 2.38i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.213 + 1.48i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-6.19 + 1.81i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (3.70 - 8.10i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-1.17 + 2.57i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-9.74 - 11.2i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-2.16 - 15.0i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (0.591 + 0.380i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-6.81 - 2.00i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-2.79 + 1.79i)T + (40.2 - 88.2i)T^{2} \) |
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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
−12.07317648170599098084971950164, −10.88848080337486464263206440364, −9.667481720408946917293548114525, −8.957907175071422657251777731809, −8.292238490492557184458866720439, −7.32134408649855081541947897615, −6.50734005082725744970230250913, −5.45426166561186146143682556840, −4.10303275734135038243060259517, −2.43438543959885597498379869715,
0.66618539066464123493352759637, 2.21611801369370950441132206462, 3.53549189619329594906589091050, 4.32456961938241557797572308294, 6.32169796422400582295983249191, 7.54071244941095547129667445844, 8.579947915907747716228501709456, 9.171494271048909653107879516862, 10.29784458457042788840668701015, 10.83116101310479829608028668539
