| L(s) = 1 | + (−0.587 + 0.809i)2-s + (−0.499 + 0.162i)3-s + (−0.309 − 0.951i)4-s + (0.834 − 2.07i)5-s + (0.162 − 0.499i)6-s − i·7-s + (0.951 + 0.309i)8-s + (−2.20 + 1.60i)9-s + (1.18 + 1.89i)10-s + (−1.59 − 1.16i)11-s + (0.308 + 0.425i)12-s + (−2.98 − 4.10i)13-s + (0.809 + 0.587i)14-s + (−0.0803 + 1.17i)15-s + (−0.809 + 0.587i)16-s + (−4.26 − 1.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 + 0.572i)2-s + (−0.288 + 0.0937i)3-s + (−0.154 − 0.475i)4-s + (0.373 − 0.927i)5-s + (0.0663 − 0.204i)6-s − 0.377i·7-s + (0.336 + 0.109i)8-s + (−0.734 + 0.533i)9-s + (0.375 + 0.599i)10-s + (−0.481 − 0.350i)11-s + (0.0891 + 0.122i)12-s + (−0.826 − 1.13i)13-s + (0.216 + 0.157i)14-s + (−0.0207 + 0.302i)15-s + (−0.202 + 0.146i)16-s + (−1.03 − 0.335i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0516 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0516 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.464232 - 0.440848i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.464232 - 0.440848i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.587 - 0.809i)T \) |
| 5 | \( 1 + (-0.834 + 2.07i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + (0.499 - 0.162i)T + (2.42 - 1.76i)T^{2} \) |
| 11 | \( 1 + (1.59 + 1.16i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.98 + 4.10i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (4.26 + 1.38i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.249 + 0.766i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.10 + 5.65i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.69 - 8.29i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.36 + 4.20i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.64 - 3.64i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.28 + 3.11i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 + (1.03 - 0.335i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (13.2 - 4.31i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (3.31 - 2.41i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.61 - 3.35i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (5.31 + 1.72i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.01 - 6.20i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.11 + 8.41i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (0.870 + 2.67i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-12.0 - 3.92i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.15 - 5.19i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (7.40 - 2.40i)T + (78.4 - 57.0i)T^{2} \) |
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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
−10.92557990478774052917752833474, −10.40913530705794714825951648689, −9.217082744897493837799663031833, −8.470523615424787060843155647895, −7.62711358767147760272196765200, −6.38648135230796228715131252529, −5.26338580957370998317679286619, −4.75365674704292991140005028986, −2.61435528167803852027847884469, −0.49909114317346308457254653359,
2.08741939484853040239566089537, 3.10790561515641215535982030700, 4.66097284986578919056428843301, 6.07337201286638248570868462650, 6.90032036214419867353969252139, 8.012389614068663038926806473023, 9.311325104687250808512357396585, 9.732123109672967385810497970846, 11.05516737447857026305883792818, 11.41537552180124077034004817768
