L(s) = 1 | + (−0.0689 + 4.99i)5-s + (−2.42 + 1.39i)7-s + (15.7 − 9.07i)11-s + (−19.9 − 11.5i)13-s − 5.72·17-s + 23.1·19-s + (0.135 − 0.235i)23-s + (−24.9 − 0.689i)25-s + (−34.4 + 19.8i)29-s + (−23.6 + 41.0i)31-s + (−6.82 − 12.2i)35-s + 34.8i·37-s + (11.4 + 6.63i)41-s + (40.4 − 23.3i)43-s + (−20.4 − 35.4i)47-s + ⋯ |
L(s) = 1 | + (−0.0137 + 0.999i)5-s + (−0.345 + 0.199i)7-s + (1.42 − 0.824i)11-s + (−1.53 − 0.885i)13-s − 0.336·17-s + 1.22·19-s + (0.00590 − 0.0102i)23-s + (−0.999 − 0.0275i)25-s + (−1.18 + 0.686i)29-s + (−0.764 + 1.32i)31-s + (−0.194 − 0.348i)35-s + 0.942i·37-s + (0.280 + 0.161i)41-s + (0.940 − 0.543i)43-s + (−0.435 − 0.753i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2325613203\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2325613203\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.0689 - 4.99i)T \) |
good | 7 | \( 1 + (2.42 - 1.39i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-15.7 + 9.07i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (19.9 + 11.5i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 5.72T + 289T^{2} \) |
| 19 | \( 1 - 23.1T + 361T^{2} \) |
| 23 | \( 1 + (-0.135 + 0.235i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (34.4 - 19.8i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (23.6 - 41.0i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 34.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-11.4 - 6.63i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-40.4 + 23.3i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (20.4 + 35.4i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 91.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (68.2 + 39.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (15.5 + 27.0i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (5.98 + 3.45i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 81.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (31.7 + 55.0i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (0.142 + 0.246i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 28.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-80.4 + 46.4i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.216595953303675981169089569404, −7.923123547290586129036204998387, −7.25545131222055051784640939306, −6.53572800751011990011836626174, −5.74077372675741179240446198574, −4.79718801653161513700560419783, −3.36722113072823609771893214636, −3.10991942238344978469552768867, −1.66930903493941138148208592695, −0.06154048703490492657785609514,
1.35536114569284174538826150344, 2.32878465329913797451133335443, 3.91562470012671706686696601814, 4.40176204167174072663497225790, 5.34213274461362777608563323532, 6.30930990579573350293197060319, 7.29948515515684926439625340049, 7.70200072808818358326383844579, 9.151439026886567064464627887519, 9.411364173467277305368066219134