L(s) = 1 | + (−1.44 + 1.95i)2-s + (2.08 − 0.395i)3-s + (0.619 + 2.00i)4-s + (2.29 − 10.9i)5-s + (−2.23 + 4.65i)6-s + (6.13 + 17.4i)7-s + (−23.1 − 8.10i)8-s + (−20.9 + 8.21i)9-s + (18.0 + 20.2i)10-s + (8.20 − 20.8i)11-s + (2.08 + 3.94i)12-s + (−33.4 − 3.77i)13-s + (−42.9 − 13.2i)14-s + (0.458 − 23.7i)15-s + (35.3 − 24.0i)16-s + (−5.07 − 0.189i)17-s + ⋯ |
L(s) = 1 | + (−0.509 + 0.690i)2-s + (0.401 − 0.0760i)3-s + (0.0774 + 0.250i)4-s + (0.204 − 0.978i)5-s + (−0.152 + 0.316i)6-s + (0.331 + 0.943i)7-s + (−1.02 − 0.358i)8-s + (−0.775 + 0.304i)9-s + (0.571 + 0.640i)10-s + (0.224 − 0.572i)11-s + (0.0501 + 0.0949i)12-s + (−0.714 − 0.0804i)13-s + (−0.820 − 0.252i)14-s + (0.00789 − 0.408i)15-s + (0.552 − 0.376i)16-s + (−0.0724 − 0.00271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0565575 - 0.584442i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0565575 - 0.584442i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.29 + 10.9i)T \) |
| 7 | \( 1 + (-6.13 - 17.4i)T \) |
good | 2 | \( 1 + (1.44 - 1.95i)T + (-2.35 - 7.64i)T^{2} \) |
| 3 | \( 1 + (-2.08 + 0.395i)T + (25.1 - 9.86i)T^{2} \) |
| 11 | \( 1 + (-8.20 + 20.8i)T + (-975. - 905. i)T^{2} \) |
| 13 | \( 1 + (33.4 + 3.77i)T + (2.14e3 + 488. i)T^{2} \) |
| 17 | \( 1 + (5.07 + 0.189i)T + (4.89e3 + 367. i)T^{2} \) |
| 19 | \( 1 + (67.0 - 116. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-6.33 - 169. i)T + (-1.21e4 + 909. i)T^{2} \) |
| 29 | \( 1 + (-38.5 + 8.78i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (141. - 81.7i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-46.4 + 24.5i)T + (2.85e4 - 4.18e4i)T^{2} \) |
| 41 | \( 1 + (195. + 405. i)T + (-4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-76.0 - 217. i)T + (-6.21e4 + 4.95e4i)T^{2} \) |
| 47 | \( 1 + (416. + 307. i)T + (3.06e4 + 9.92e4i)T^{2} \) |
| 53 | \( 1 + (-94.6 - 50.0i)T + (8.38e4 + 1.23e5i)T^{2} \) |
| 59 | \( 1 + (-18.1 - 242. i)T + (-2.03e5 + 3.06e4i)T^{2} \) |
| 61 | \( 1 + (-128. + 417. i)T + (-1.87e5 - 1.27e5i)T^{2} \) |
| 67 | \( 1 + (41.6 + 11.1i)T + (2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (172. - 755. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-313. + 231. i)T + (1.14e5 - 3.71e5i)T^{2} \) |
| 79 | \( 1 + (945. + 545. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-120. - 1.06e3i)T + (-5.57e5 + 1.27e5i)T^{2} \) |
| 89 | \( 1 + (102. + 261. i)T + (-5.16e5 + 4.79e5i)T^{2} \) |
| 97 | \( 1 + (-273. - 273. i)T + 9.12e5iT^{2} \) |
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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
−12.16846074030846033280140382886, −11.44256257330757793883857911550, −9.774611573721905271630572407021, −8.810182159086573279235676497684, −8.424156436036734562092964770188, −7.55842186697588759757543507249, −6.01016471266330920057062903573, −5.30725257215911531702781494940, −3.49502409045460130201839730858, −2.00672741920883706352068224945,
0.24020906309577075300268363224, 2.10196908912398110988066202060, 3.03995955331350197550997984280, 4.61537952169377828177650227053, 6.27281756254431151787452264613, 7.09413426033801329587522939852, 8.429828083447089848057916368310, 9.471463411918638809218322846924, 10.23284008259105639812792860906, 11.02938358266242065633879711254