| L(s) = 1 | + (−15.4 + 4.07i)2-s + (127. + 41.5i)3-s + (222. − 125. i)4-s + (−607. + 146. i)5-s + (−2.14e3 − 122. i)6-s + 3.59e3i·7-s + (−2.93e3 + 2.85e3i)8-s + (9.33e3 + 6.78e3i)9-s + (8.80e3 − 4.73e3i)10-s + (1.00e4 + 1.38e4i)11-s + (3.37e4 − 6.85e3i)12-s + (2.58e3 + 1.87e3i)13-s + (−1.46e4 − 5.56e4i)14-s + (−8.38e4 − 6.53e3i)15-s + (3.37e4 − 5.61e4i)16-s + (1.84e4 + 5.69e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.967 + 0.254i)2-s + (1.57 + 0.513i)3-s + (0.870 − 0.492i)4-s + (−0.972 + 0.234i)5-s + (−1.65 − 0.0944i)6-s + 1.49i·7-s + (−0.716 + 0.697i)8-s + (1.42 + 1.03i)9-s + (0.880 − 0.473i)10-s + (0.688 + 0.948i)11-s + (1.62 − 0.330i)12-s + (0.0905 + 0.0658i)13-s + (−0.381 − 1.44i)14-s + (−1.65 − 0.129i)15-s + (0.515 − 0.856i)16-s + (0.221 + 0.681i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.112076 + 1.55839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112076 + 1.55839i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (15.4 - 4.07i)T \) |
| 5 | \( 1 + (607. - 146. i)T \) |
| good | 3 | \( 1 + (-127. - 41.5i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 - 3.59e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-1.00e4 - 1.38e4i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (-2.58e3 - 1.87e3i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-1.84e4 - 5.69e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (6.29e4 - 2.04e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (1.76e5 + 2.42e5i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (-3.59e5 + 1.10e6i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (7.19e5 - 2.33e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-5.25e5 - 3.81e5i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-1.75e5 - 1.27e5i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 - 6.23e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (4.22e6 + 1.37e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (1.06e6 - 3.27e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (8.44e6 - 1.16e7i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-1.67e7 + 1.21e7i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (2.64e6 - 8.59e5i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (3.78e7 + 1.23e7i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (2.52e7 - 1.83e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (-5.02e7 - 1.63e7i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (2.66e7 - 8.65e6i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-6.90e7 + 5.01e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (2.46e7 - 7.57e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.56477109922437121852593624575, −11.62407904393413038073199831877, −10.16940241284219511660601831122, −9.259840597351345596817861689519, −8.442248725187888332001081380888, −7.82327000157389141101512780537, −6.35732736638881093706181342966, −4.31186235374283879124261073230, −2.86723730778312349470862910143, −1.90403921770748747365490197682,
0.49421617845567336235883511979, 1.49058782593076081041261872407, 3.24695173883617893162142231533, 3.85305394038952302337107163318, 6.93409831240251562411308491735, 7.55388048067295552974607386572, 8.443199758796780958525517088778, 9.232101649518805976336765299551, 10.55487279020068120917759817036, 11.63004562185437756260133466374
