| L(s) = 1 | + (−5.74 − 6.85i)3-s + (55.2 + 20.1i)5-s + (−190. + 329. i)7-s + (112. − 639. i)9-s + (−520. − 902. i)11-s + (−159. + 190. i)13-s + (−179. − 494. i)15-s + (967. + 5.48e3i)17-s + (−6.84e3 − 451. i)19-s + (3.35e3 − 591. i)21-s + (−1.90e4 + 6.93e3i)23-s + (−9.31e3 − 7.81e3i)25-s + (−1.06e4 + 6.16e3i)27-s + (−3.55e4 − 6.26e3i)29-s + (2.40e4 + 1.39e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.212 − 0.253i)3-s + (0.442 + 0.160i)5-s + (−0.554 + 0.961i)7-s + (0.154 − 0.876i)9-s + (−0.391 − 0.677i)11-s + (−0.0727 + 0.0866i)13-s + (−0.0533 − 0.146i)15-s + (0.196 + 1.11i)17-s + (−0.997 − 0.0657i)19-s + (0.361 − 0.0638i)21-s + (−1.56 + 0.570i)23-s + (−0.596 − 0.500i)25-s + (−0.542 + 0.313i)27-s + (−1.45 − 0.256i)29-s + (0.808 + 0.466i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0135768 + 0.124539i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0135768 + 0.124539i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (6.84e3 + 451. i)T \) |
| good | 3 | \( 1 + (5.74 + 6.85i)T + (-126. + 717. i)T^{2} \) |
| 5 | \( 1 + (-55.2 - 20.1i)T + (1.19e4 + 1.00e4i)T^{2} \) |
| 7 | \( 1 + (190. - 329. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (520. + 902. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (159. - 190. i)T + (-8.38e5 - 4.75e6i)T^{2} \) |
| 17 | \( 1 + (-967. - 5.48e3i)T + (-2.26e7 + 8.25e6i)T^{2} \) |
| 23 | \( 1 + (1.90e4 - 6.93e3i)T + (1.13e8 - 9.51e7i)T^{2} \) |
| 29 | \( 1 + (3.55e4 + 6.26e3i)T + (5.58e8 + 2.03e8i)T^{2} \) |
| 31 | \( 1 + (-2.40e4 - 1.39e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 7.88e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (4.69e4 + 5.59e4i)T + (-8.24e8 + 4.67e9i)T^{2} \) |
| 43 | \( 1 + (7.70e4 + 2.80e4i)T + (4.84e9 + 4.06e9i)T^{2} \) |
| 47 | \( 1 + (2.06e4 - 1.17e5i)T + (-1.01e10 - 3.68e9i)T^{2} \) |
| 53 | \( 1 + (-4.53e4 - 1.24e5i)T + (-1.69e10 + 1.42e10i)T^{2} \) |
| 59 | \( 1 + (-1.65e5 + 2.91e4i)T + (3.96e10 - 1.44e10i)T^{2} \) |
| 61 | \( 1 + (-2.76e5 + 1.00e5i)T + (3.94e10 - 3.31e10i)T^{2} \) |
| 67 | \( 1 + (5.07e5 + 8.95e4i)T + (8.50e10 + 3.09e10i)T^{2} \) |
| 71 | \( 1 + (-1.89e5 + 5.19e5i)T + (-9.81e10 - 8.23e10i)T^{2} \) |
| 73 | \( 1 + (-4.17e5 + 3.50e5i)T + (2.62e10 - 1.49e11i)T^{2} \) |
| 79 | \( 1 + (-3.52e5 - 4.19e5i)T + (-4.22e10 + 2.39e11i)T^{2} \) |
| 83 | \( 1 + (1.67e5 - 2.89e5i)T + (-1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + (-5.81e5 + 6.92e5i)T + (-8.62e10 - 4.89e11i)T^{2} \) |
| 97 | \( 1 + (-4.34e5 + 7.65e4i)T + (7.82e11 - 2.84e11i)T^{2} \) |
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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
−13.66917676344833497114032652579, −12.62206464395081107462354139491, −11.83107259855631443502551889287, −10.39444635997065827058836356543, −9.327923419510936135071086938130, −8.153820318272099759952016127177, −6.36991796635429629398062609216, −5.79622618839473170342204095778, −3.66574410733139587810033894596, −2.01917483198868792928040901075,
0.04473060627778211099068596972, 2.10783926080665875048278049818, 4.07881086902714801395633704761, 5.34864032090611184253634682137, 6.87860562943139425294292672250, 8.028719503855979973717123219608, 9.806416133295467268939440535704, 10.26504397446304137609852787263, 11.60439325379219786091965708879, 13.07501925968978175033147867936
