L(s) = 1 | + (−0.309 + 0.951i)2-s + (0.5 − 0.363i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.190 + 0.587i)6-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 2.48i)9-s + 0.999·10-s + (0.809 + 3.21i)11-s − 0.618·12-s + (−0.236 + 0.726i)13-s + (0.809 − 0.587i)14-s + (−0.5 − 0.363i)15-s + (0.309 + 0.951i)16-s + (0.118 + 0.363i)17-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.672i)2-s + (0.288 − 0.209i)3-s + (−0.404 − 0.293i)4-s + (−0.138 − 0.425i)5-s + (0.0779 + 0.239i)6-s + (−0.305 − 0.222i)7-s + (0.286 − 0.207i)8-s + (−0.269 + 0.829i)9-s + 0.316·10-s + (0.243 + 0.969i)11-s − 0.178·12-s + (−0.0654 + 0.201i)13-s + (0.216 − 0.157i)14-s + (−0.129 − 0.0937i)15-s + (0.0772 + 0.237i)16-s + (0.0286 + 0.0881i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.877352 + 0.858272i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.877352 + 0.858272i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
| 7 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 + (-0.809 - 3.21i)T \) |
good | 3 | \( 1 + (-0.5 + 0.363i)T + (0.927 - 2.85i)T^{2} \) |
| 13 | \( 1 + (0.236 - 0.726i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.118 - 0.363i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.927 - 0.673i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 + (-3.61 - 2.62i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2 - 6.15i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.47 - 3.97i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.11 + 2.26i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 9.09T + 43T^{2} \) |
| 47 | \( 1 + (7.47 - 5.42i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (2.23 - 6.88i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (8.78 + 6.37i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.0901 - 0.277i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 5.61T + 67T^{2} \) |
| 71 | \( 1 + (2.14 + 6.60i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.11 - 2.26i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.23 + 3.80i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.663 + 2.04i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 8.14T + 89T^{2} \) |
| 97 | \( 1 + (3.28 - 10.0i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.43911081130940035145373782820, −9.377426053313712312314355309345, −8.833155331729983036498366610301, −7.82859159137928719969075310560, −7.21871121269412303424888371351, −6.32830580614028160642863279334, −5.07642476909370383560368844732, −4.46538170804989560808534817160, −2.94760118567541109248047283790, −1.42205800234605816658568412209,
0.71150018506487859814641346662, 2.64709525694873745018113978075, 3.31163370614067613175126902385, 4.30849656905041480001551140722, 5.71602392363153508871007039461, 6.57522159051783907433885961997, 7.70671430784639042364018710944, 8.718406093385376481556229902453, 9.254820599029903243090819771316, 10.05157434325184337418049454806