| L(s) = 1 | + (−2 + 3.46i)2-s + (−3.86 + 6.68i)3-s + (−7.99 − 13.8i)4-s + (46.3 − 80.3i)5-s + (−15.4 − 26.7i)6-s + 13.5·7-s + 63.9·8-s + (91.6 + 158. i)9-s + (185. + 321. i)10-s + 407.·11-s + 123.·12-s + (−86.6 − 150. i)13-s + (−27.0 + 46.9i)14-s + (358. + 620. i)15-s + (−128 + 221. i)16-s + (822. − 1.42e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.247 + 0.429i)3-s + (−0.249 − 0.433i)4-s + (0.829 − 1.43i)5-s + (−0.175 − 0.303i)6-s + 0.104·7-s + 0.353·8-s + (0.377 + 0.653i)9-s + (0.586 + 1.01i)10-s + 1.01·11-s + 0.247·12-s + (−0.142 − 0.246i)13-s + (−0.0369 + 0.0639i)14-s + (0.410 + 0.711i)15-s + (−0.125 + 0.216i)16-s + (0.690 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.44871 + 0.141750i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44871 + 0.141750i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 - 3.46i)T \) |
| 19 | \( 1 + (-1.57e3 - 27.9i)T \) |
| good | 3 | \( 1 + (3.86 - 6.68i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (-46.3 + 80.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 - 13.5T + 1.68e4T^{2} \) |
| 11 | \( 1 - 407.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (86.6 + 150. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-822. + 1.42e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (-778. - 1.34e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (2.20e3 + 3.81e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 1.48e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.11e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (9.66e3 - 1.67e4i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-7.84e3 + 1.35e4i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-6.51e3 - 1.12e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-820. - 1.42e3i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (2.34e4 - 4.06e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.38e4 + 2.39e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (6.09e3 + 1.05e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (6.07e3 - 1.05e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (2.30e4 - 3.98e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (2.26e4 - 3.92e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 2.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (5.75e4 + 9.97e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (1.95e4 - 3.37e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−15.71914372812583369668296940453, −14.12182948817379327869309178710, −13.16910691322104839230518168844, −11.67333786209880031910790889578, −9.836512979984287251146414921046, −9.216093812196277248307554844390, −7.64844536032300105947230201479, −5.68927117932890158187300857527, −4.70771868366663079448718224737, −1.21386934800942374400245448098,
1.58961081282473049753404043500, 3.45260822281105648367370282612, 6.16752379674396624988752092214, 7.25661304167949953723065053437, 9.298514749734345819503805984329, 10.34323473321207605132207964334, 11.49491725495303207868512580161, 12.66395796914739456495128145834, 14.09171782027158295510088904513, 14.90669632568257192877313302404
