| L(s) = 1 | + (3.23 + 2.35i)2-s + (−4.67 − 14.3i)3-s + (4.94 + 15.2i)4-s + (27.5 − 48.6i)5-s + (18.7 − 57.5i)6-s − 181.·7-s + (−19.7 + 60.8i)8-s + (11.4 − 8.29i)9-s + (203. − 92.4i)10-s + (−575. − 417. i)11-s + (195. − 142. i)12-s + (694. − 504. i)13-s + (−588. − 427. i)14-s + (−828. − 169. i)15-s + (−207. + 150. i)16-s + (30.9 − 95.2i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.299 − 0.923i)3-s + (0.154 + 0.475i)4-s + (0.493 − 0.869i)5-s + (0.212 − 0.652i)6-s − 1.40·7-s + (−0.109 + 0.336i)8-s + (0.0470 − 0.0341i)9-s + (0.643 − 0.292i)10-s + (−1.43 − 1.04i)11-s + (0.392 − 0.285i)12-s + (1.13 − 0.827i)13-s + (−0.802 − 0.582i)14-s + (−0.950 − 0.194i)15-s + (−0.202 + 0.146i)16-s + (0.0259 − 0.0799i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.00917 - 1.25432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00917 - 1.25432i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 5 | \( 1 + (-27.5 + 48.6i)T \) |
| good | 3 | \( 1 + (4.67 + 14.3i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 + 181.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (575. + 417. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-694. + 504. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-30.9 + 95.2i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-140. + 433. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-3.11e3 - 2.26e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-450. - 1.38e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (1.79e3 - 5.53e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-6.98e3 + 5.07e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (581. - 422. i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.80e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-7.58e3 - 2.33e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (6.50e3 + 2.00e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-8.81e3 + 6.40e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.09e4 + 7.95e3i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-1.60e4 + 4.93e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (1.40e4 + 4.32e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-2.52e4 - 1.83e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (9.19e3 + 2.82e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-2.57e4 + 7.91e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (6.07e4 + 4.41e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-4.08e4 - 1.25e5i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−13.63462578866639038099700628833, −13.02784620281552677695093960979, −12.63674875823822174388427324185, −10.85376657960848722757019617719, −9.172756815261022760556067701354, −7.76729319908338300691600280305, −6.30207449472692205332316715251, −5.47670627780163654443112378478, −3.16036613915645179511924789095, −0.70769201833233100385403292083,
2.61717968445892610106922770424, 4.10978827400107645613513422250, 5.71732866493235200350212617747, 6.98488324847959150279791256791, 9.509211804196817314664906614840, 10.26550337108048959046473418527, 11.05513859916399687048192513250, 12.80116528886398169288408486538, 13.52349864879249999922826164770, 15.04706890991185831054862382277
