| 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 + (0.626 − 4.35i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.307 − 0.673i)11-s + (0.415 + 0.909i)12-s + (0.494 + 3.43i)13-s + (3.70 + 2.38i)14-s + (0.959 − 0.281i)15-s + (−0.142 + 0.989i)16-s + (−5.19 + 5.99i)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 + (0.236 − 1.64i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.0450 − 0.313i)10-s + (−0.0927 − 0.203i)11-s + (0.119 + 0.262i)12-s + (0.137 + 0.953i)13-s + (0.990 + 0.636i)14-s + (0.247 − 0.0727i)15-s + (−0.0355 + 0.247i)16-s + (−1.25 + 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00857499 - 0.0441995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00857499 - 0.0441995i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (3.39 + 3.38i)T \) |
| good | 7 | \( 1 + (-0.626 + 4.35i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (0.307 + 0.673i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.494 - 3.43i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (5.19 - 5.99i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (1.66 + 1.92i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (5.82 - 6.71i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.754 + 0.221i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (8.83 + 5.67i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-9.19 + 5.90i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (0.408 + 0.120i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + (-0.588 + 4.09i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.23 - 8.61i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (3.04 - 0.894i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (5.32 - 11.6i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (3.99 - 8.74i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (2.74 + 3.16i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.0190 - 0.132i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (10.4 + 6.69i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (7.81 + 2.29i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (8.12 - 5.21i)T + (40.2 - 88.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.46524319843393637244491723633, −9.043203930926152261985486512860, −8.257552289994087477370960656195, −7.11558158069932458880733029821, −6.88757060632871660851828815436, −5.81030898023323978915971613452, −4.38473189116547551950584835134, −3.97882085248193928610320544211, −1.69346224050113056798985276801, −0.02746044117016939803936361084,
1.96451811411991632060170017239, 3.08308965408461517752556087929, 4.49580860252822450660532959735, 5.32604096403042442115872024590, 6.21442535397086271310008444016, 7.61416060400972260249907033477, 8.415827336121585698887279026439, 9.266852666551696328061130557895, 9.903298849467210645173827527162, 11.13346151648996614897946323453
