| L(s) = 1 | + (35.0 − 30.9i)3-s + 125i·5-s + 201. i·7-s + (267. − 2.17e3i)9-s − 499.·11-s + 1.35e3·13-s + (3.87e3 + 4.37e3i)15-s − 1.67e3i·17-s − 1.61e4i·19-s + (6.23e3 + 7.05e3i)21-s − 1.68e4·23-s − 1.56e4·25-s + (−5.78e4 − 8.43e4i)27-s + 8.26e4i·29-s − 1.61e5i·31-s + ⋯ |
| L(s) = 1 | + (0.749 − 0.662i)3-s + 0.447i·5-s + 0.221i·7-s + (0.122 − 0.992i)9-s − 0.113·11-s + 0.171·13-s + (0.296 + 0.335i)15-s − 0.0824i·17-s − 0.539i·19-s + (0.146 + 0.166i)21-s − 0.288·23-s − 0.199·25-s + (−0.565 − 0.824i)27-s + 0.628i·29-s − 0.975i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.182160762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.182160762\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-35.0 + 30.9i)T \) |
| 5 | \( 1 - 125iT \) |
| good | 7 | \( 1 - 201. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 499.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.35e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.67e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 1.61e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 1.68e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 8.26e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.61e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 1.28e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.74e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + 7.66e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 5.58e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.13e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 8.80e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 7.56e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.60e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 1.92e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.17e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.22e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 1.39e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.73e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 1.25e7T + 8.07e13T^{2} \) |
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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
−10.56963881311874958432013967061, −9.439586976093549337596965859836, −8.601185842616798064446427748045, −7.57631741490184507205491738911, −6.76406798340059992355984645633, −5.65274490385765341671781317863, −4.02763295782811136151955882526, −2.88339855313704585410182278525, −1.90898198435224309928038010046, −0.45935378467578438783206655509,
1.29069990088765852215503048955, 2.66039421197466712649565726743, 3.86886562257035419539623235744, 4.76527213165705078194720953847, 5.98847455665653116798337495584, 7.50379197495358909229073653190, 8.326066872531213868968729094637, 9.233956970089979250486431149076, 10.09800345270737123670532323966, 10.95178400766299558999932061600
