L(s) = 1 | + (0.381 − 1.36i)2-s + (−0.605 + 0.605i)3-s + (−1.70 − 1.03i)4-s + (−0.707 − 0.707i)5-s + (0.593 + 1.05i)6-s + i·7-s + (−2.06 + 1.93i)8-s + 2.26i·9-s + (−1.23 + 0.693i)10-s + (−0.812 − 0.812i)11-s + (1.66 − 0.406i)12-s + (−2.79 + 2.79i)13-s + (1.36 + 0.381i)14-s + 0.856·15-s + (1.84 + 3.54i)16-s + 4.69·17-s + ⋯ |
L(s) = 1 | + (0.269 − 0.963i)2-s + (−0.349 + 0.349i)3-s + (−0.854 − 0.519i)4-s + (−0.316 − 0.316i)5-s + (0.242 + 0.430i)6-s + 0.377i·7-s + (−0.730 + 0.683i)8-s + 0.755i·9-s + (−0.389 + 0.219i)10-s + (−0.244 − 0.244i)11-s + (0.480 − 0.117i)12-s + (−0.776 + 0.776i)13-s + (0.363 + 0.101i)14-s + 0.221·15-s + (0.461 + 0.887i)16-s + 1.13·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.724115 + 0.337405i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.724115 + 0.337405i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.381 + 1.36i)T \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (0.605 - 0.605i)T - 3iT^{2} \) |
| 11 | \( 1 + (0.812 + 0.812i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.79 - 2.79i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.69T + 17T^{2} \) |
| 19 | \( 1 + (1.00 - 1.00i)T - 19iT^{2} \) |
| 23 | \( 1 - 8.15iT - 23T^{2} \) |
| 29 | \( 1 + (-0.122 + 0.122i)T - 29iT^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 + (-3.53 - 3.53i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.84iT - 41T^{2} \) |
| 43 | \( 1 + (2.21 + 2.21i)T + 43iT^{2} \) |
| 47 | \( 1 + 3.94T + 47T^{2} \) |
| 53 | \( 1 + (-7.19 - 7.19i)T + 53iT^{2} \) |
| 59 | \( 1 + (6.54 + 6.54i)T + 59iT^{2} \) |
| 61 | \( 1 + (1.81 - 1.81i)T - 61iT^{2} \) |
| 67 | \( 1 + (0.162 - 0.162i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.00iT - 71T^{2} \) |
| 73 | \( 1 + 7.09iT - 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + (7.83 - 7.83i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.28iT - 89T^{2} \) |
| 97 | \( 1 - 0.453T + 97T^{2} \) |
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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
−11.07581282196856245600235115013, −10.01568868307944367052295589450, −9.511558864138529308971935540364, −8.390935132005203462136760429544, −7.49987204870574676012341743067, −5.82151561660824562395508813753, −5.14497838833742214050041167132, −4.25325654132209162777612211136, −3.05281392516932651761463488111, −1.67779417830703602922374466059,
0.44580874989943230416913554391, 2.98589451265163655407544724780, 4.12070458014699830066484152310, 5.22893881697755431915718655688, 6.17543070283818441871598444031, 7.05784793719406064069707633053, 7.66397413083333907697105095100, 8.615180575093824826484049784392, 9.738904301902015825494983842217, 10.50511319359516317621135054758