| L(s) = 1 | + (−2.10 + 1.88i)2-s + (−3.74 + 3.60i)3-s + (0.868 − 7.95i)4-s + (−1.60 − 4.41i)5-s + (1.07 − 14.6i)6-s + (23.1 − 4.07i)7-s + (13.1 + 18.3i)8-s + (1.00 − 26.9i)9-s + (11.7 + 6.25i)10-s + (−26.1 − 9.51i)11-s + (25.4 + 32.8i)12-s + (58.1 + 48.8i)13-s + (−41.0 + 52.2i)14-s + (21.9 + 10.7i)15-s + (−62.4 − 13.8i)16-s + (21.6 + 12.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.744 + 0.667i)2-s + (−0.720 + 0.693i)3-s + (0.108 − 0.994i)4-s + (−0.143 − 0.394i)5-s + (0.0728 − 0.997i)6-s + (1.24 − 0.220i)7-s + (0.582 + 0.812i)8-s + (0.0370 − 0.999i)9-s + (0.370 + 0.197i)10-s + (−0.716 − 0.260i)11-s + (0.611 + 0.791i)12-s + (1.24 + 1.04i)13-s + (−0.782 + 0.997i)14-s + (0.377 + 0.184i)15-s + (−0.976 − 0.215i)16-s + (0.309 + 0.178i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.777944 + 0.533821i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.777944 + 0.533821i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.10 - 1.88i)T \) |
| 3 | \( 1 + (3.74 - 3.60i)T \) |
| good | 5 | \( 1 + (1.60 + 4.41i)T + (-95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-23.1 + 4.07i)T + (322. - 117. i)T^{2} \) |
| 11 | \( 1 + (26.1 + 9.51i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (-58.1 - 48.8i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-21.6 - 12.5i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (39.2 - 22.6i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (13.9 - 79.2i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-167. - 199. i)T + (-4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-246. - 43.4i)T + (2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-70.2 + 121. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-19.4 + 23.1i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-41.0 + 112. i)T + (-6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (81.7 + 463. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 - 660. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-363. + 132. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (43.2 + 245. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (394. - 469. i)T + (-5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-145. + 252. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-302. - 523. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-81.0 - 96.6i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-676. + 567. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (5.70 - 3.29i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (634. + 230. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−13.76971873442129072888328426201, −11.98379238843860515904642090866, −10.99504697049406550450179759485, −10.36591366051696521453544435871, −8.901741735984548252217763150497, −8.178196262231903668204143378871, −6.63427879109314327198075949123, −5.39887197161209375609427289612, −4.38630273589306912199428598202, −1.17536528016810418036730413131,
0.967214092730377923128989889553, 2.59432507385150504952300692089, 4.75968650212531480229945537952, 6.39129900353778980165497006469, 7.897158882813914512444098208018, 8.279543477247487107873695589140, 10.25233983657419865995174850844, 10.98708321623054977826265369053, 11.69874188399552947539277123653, 12.77133032471638116829806623013
