| L(s) = 1 | + (2.10 + 1.21i)2-s + (−3.73 − 6.46i)3-s + (−5.05 − 8.75i)4-s + (24.9 − 0.969i)5-s − 18.1i·6-s + (−22.9 − 43.3i)7-s − 63.3i·8-s + (12.6 − 21.8i)9-s + (53.6 + 28.2i)10-s + (95.0 + 164. i)11-s + (−37.7 + 65.3i)12-s − 63.2·13-s + (4.42 − 118. i)14-s + (−99.5 − 157. i)15-s + (−3.93 + 6.81i)16-s + (166. + 288. i)17-s + ⋯ |
| L(s) = 1 | + (0.525 + 0.303i)2-s + (−0.414 − 0.718i)3-s + (−0.315 − 0.547i)4-s + (0.999 − 0.0387i)5-s − 0.503i·6-s + (−0.467 − 0.884i)7-s − 0.990i·8-s + (0.155 − 0.269i)9-s + (0.536 + 0.282i)10-s + (0.785 + 1.36i)11-s + (−0.262 + 0.453i)12-s − 0.374·13-s + (0.0225 − 0.606i)14-s + (−0.442 − 0.701i)15-s + (−0.0153 + 0.0266i)16-s + (0.575 + 0.996i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.38106 - 0.917081i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38106 - 0.917081i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-24.9 + 0.969i)T \) |
| 7 | \( 1 + (22.9 + 43.3i)T \) |
| good | 2 | \( 1 + (-2.10 - 1.21i)T + (8 + 13.8i)T^{2} \) |
| 3 | \( 1 + (3.73 + 6.46i)T + (-40.5 + 70.1i)T^{2} \) |
| 11 | \( 1 + (-95.0 - 164. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + 63.2T + 2.85e4T^{2} \) |
| 17 | \( 1 + (-166. - 288. i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (343. + 198. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-361. - 208. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 - 1.38e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + (-416. + 240. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-1.36e3 - 788. i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + 959. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.68e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (868. - 1.50e3i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (2.06e3 - 1.19e3i)T + (3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-131. + 76.0i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (2.35e3 + 1.35e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (2.68e3 - 1.55e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 739.T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-2.97e3 - 5.14e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-1.71e3 + 2.97e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 1.09e4T + 4.74e7T^{2} \) |
| 89 | \( 1 + (2.30e3 + 1.32e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 5.79e3T + 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.24030646332719122527305118456, −14.26104019518873256644486440368, −13.13128372727279307389923576969, −12.45757524276265878438991888722, −10.32995797954623831131681505419, −9.486603127066566250854368793315, −6.93989829092536929402857318920, −6.24177634367518227046259212571, −4.46559872925799341752749758531, −1.24977980630715878777546876687,
2.91371132365876966185024691474, 4.83254019556949706795551885020, 6.09928904405309615109305698626, 8.591145767454780082730835820035, 9.755139342935797916854682715984, 11.19793026217906361828503814059, 12.41028411518362126154898509546, 13.56967334490943708880807889087, 14.56815980879369491058878665088, 16.31535057801916794923150535398
