| L(s) = 1 | + (0.415 + 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (−0.654 − 0.755i)6-s + (−0.498 − 3.46i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (−0.142 + 0.989i)10-s + (1.29 − 2.84i)11-s + (0.415 − 0.909i)12-s + (−0.133 + 0.930i)13-s + (2.94 − 1.89i)14-s + (−0.959 − 0.281i)15-s + (−0.142 − 0.989i)16-s + (−4.81 − 5.56i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (0.376 + 0.241i)5-s + (−0.267 − 0.308i)6-s + (−0.188 − 1.30i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.0450 + 0.313i)10-s + (0.391 − 0.857i)11-s + (0.119 − 0.262i)12-s + (−0.0371 + 0.258i)13-s + (0.786 − 0.505i)14-s + (−0.247 − 0.0727i)15-s + (−0.0355 − 0.247i)16-s + (−1.16 − 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26897 - 0.251675i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26897 - 0.251675i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 5 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (-3.42 + 3.35i)T \) |
| good | 7 | \( 1 + (0.498 + 3.46i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 2.84i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.133 - 0.930i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (4.81 + 5.56i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-1.23 + 1.42i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (0.455 + 0.525i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-4.98 - 1.46i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (0.958 - 0.615i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-1.73 - 1.11i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-9.55 + 2.80i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 3.43T + 47T^{2} \) |
| 53 | \( 1 + (0.573 + 3.98i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (1.82 - 12.7i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (2.91 + 0.854i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-1.53 - 3.35i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (1.92 + 4.20i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (0.931 - 1.07i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-2.40 + 16.7i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-5.18 + 3.32i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (3.33 - 0.979i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (4.25 + 2.73i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.53733087552777706490352433987, −9.493826555244796795561971661062, −8.753392502084128537743546734581, −7.39982278301844518134991635429, −6.82552554143344335678350144444, −6.11222870811911575389660965301, −4.90851133208064224188845117761, −4.17659404539778015041688984783, −2.92448314154834532014461455438, −0.69964990870935352868308967651,
1.58248988108016105213600443754, 2.58007897373578894046254865556, 4.08012049480695976077926445896, 5.10724278536392837199520678524, 5.91179871621247842179080703452, 6.66399392287795518239295775293, 8.076551431613354296142375863445, 9.103796580841197014041435900369, 9.643938552967243114454012038389, 10.68065827658672881186635220517
