| L(s) = 1 | + (2.69 − 15.2i)3-s + (0.287 + 0.241i)5-s + (−83.3 − 144. i)7-s + (2.83 + 1.03i)9-s + (−246. + 426. i)11-s + (−182. − 1.03e3i)13-s + (4.45 − 3.73i)15-s + (−1.05e3 + 383. i)17-s + (132. + 1.56e3i)19-s + (−2.42e3 + 883. i)21-s + (−2.62e3 + 2.20e3i)23-s + (−542. − 3.07e3i)25-s + (1.90e3 − 3.30e3i)27-s + (−5.23e3 − 1.90e3i)29-s + (−2.62e3 − 4.54e3i)31-s + ⋯ |
| L(s) = 1 | + (0.172 − 0.978i)3-s + (0.00514 + 0.00431i)5-s + (−0.642 − 1.11i)7-s + (0.0116 + 0.00424i)9-s + (−0.613 + 1.06i)11-s + (−0.299 − 1.69i)13-s + (0.00511 − 0.00428i)15-s + (−0.883 + 0.321i)17-s + (0.0844 + 0.996i)19-s + (−1.20 + 0.437i)21-s + (−1.03 + 0.868i)23-s + (−0.173 − 0.984i)25-s + (0.503 − 0.871i)27-s + (−1.15 − 0.421i)29-s + (−0.490 − 0.848i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.123i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0575338 - 0.930099i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0575338 - 0.930099i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-132. - 1.56e3i)T \) |
| good | 3 | \( 1 + (-2.69 + 15.2i)T + (-228. - 83.1i)T^{2} \) |
| 5 | \( 1 + (-0.287 - 0.241i)T + (542. + 3.07e3i)T^{2} \) |
| 7 | \( 1 + (83.3 + 144. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (246. - 426. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (182. + 1.03e3i)T + (-3.48e5 + 1.26e5i)T^{2} \) |
| 17 | \( 1 + (1.05e3 - 383. i)T + (1.08e6 - 9.12e5i)T^{2} \) |
| 23 | \( 1 + (2.62e3 - 2.20e3i)T + (1.11e6 - 6.33e6i)T^{2} \) |
| 29 | \( 1 + (5.23e3 + 1.90e3i)T + (1.57e7 + 1.31e7i)T^{2} \) |
| 31 | \( 1 + (2.62e3 + 4.54e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 662.T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-1.94e3 + 1.10e4i)T + (-1.08e8 - 3.96e7i)T^{2} \) |
| 43 | \( 1 + (-1.19e4 - 1.00e4i)T + (2.55e7 + 1.44e8i)T^{2} \) |
| 47 | \( 1 + (-2.30e4 - 8.37e3i)T + (1.75e8 + 1.47e8i)T^{2} \) |
| 53 | \( 1 + (-1.76e4 + 1.47e4i)T + (7.26e7 - 4.11e8i)T^{2} \) |
| 59 | \( 1 + (1.13e4 - 4.14e3i)T + (5.47e8 - 4.59e8i)T^{2} \) |
| 61 | \( 1 + (-2.49e4 + 2.09e4i)T + (1.46e8 - 8.31e8i)T^{2} \) |
| 67 | \( 1 + (1.75e4 + 6.38e3i)T + (1.03e9 + 8.67e8i)T^{2} \) |
| 71 | \( 1 + (5.87e3 + 4.93e3i)T + (3.13e8 + 1.77e9i)T^{2} \) |
| 73 | \( 1 + (-4.77e3 + 2.70e4i)T + (-1.94e9 - 7.09e8i)T^{2} \) |
| 79 | \( 1 + (-1.54e4 + 8.75e4i)T + (-2.89e9 - 1.05e9i)T^{2} \) |
| 83 | \( 1 + (1.40e4 + 2.43e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-1.31e4 - 7.46e4i)T + (-5.24e9 + 1.90e9i)T^{2} \) |
| 97 | \( 1 + (-6.78e4 + 2.46e4i)T + (6.57e9 - 5.51e9i)T^{2} \) |
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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
−12.98891011482799314562448452340, −12.36036480934440415130012188717, −10.56149623745099164167658805258, −9.850629678379265484002147942903, −7.81207897090105559092813411527, −7.41470149073951977238033683260, −5.93363757279673344567811214807, −4.02270554456917474593748308217, −2.15185010796304684704184977981, −0.37254037649788067851118240024,
2.53488320791999382587535783632, 4.07259514418658768993791716450, 5.48000725811782680103339991109, 6.89987480698149592436620208495, 8.955844671005936518636659087109, 9.228049016369194390296127227728, 10.69377504293219238965237612348, 11.72659615273613754051198324896, 13.00171212281877444609821750252, 14.15774592345595879320030596296
