| L(s) = 1 | + (−3.23 + 2.35i)2-s + (2.78 − 8.55i)3-s + (4.94 − 15.2i)4-s + (−8.34 + 55.2i)5-s + (11.1 + 34.2i)6-s − 188.·7-s + (19.7 + 60.8i)8-s + (−65.5 − 47.6i)9-s + (−102. − 198. i)10-s + (115. − 83.6i)11-s + (−116. − 84.6i)12-s + (282. + 205. i)13-s + (610. − 443. i)14-s + (449. + 225. i)15-s + (−207. − 150. i)16-s + (210. + 646. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (−0.149 + 0.988i)5-s + (0.126 + 0.388i)6-s − 1.45·7-s + (0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + (−0.325 − 0.627i)10-s + (0.286 − 0.208i)11-s + (−0.233 − 0.169i)12-s + (0.463 + 0.336i)13-s + (0.831 − 0.604i)14-s + (0.516 + 0.258i)15-s + (−0.202 − 0.146i)16-s + (0.176 + 0.542i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9293562204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9293562204\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 - 2.35i)T \) |
| 3 | \( 1 + (-2.78 + 8.55i)T \) |
| 5 | \( 1 + (8.34 - 55.2i)T \) |
| good | 7 | \( 1 + 188.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-115. + 83.6i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-282. - 205. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-210. - 646. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (216. + 665. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.18e3 + 860. i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-2.10e3 + 6.46e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (676. + 2.08e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-1.30e3 - 949. i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (5.08e3 + 3.69e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 2.34e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-3.28e3 + 1.01e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-5.48e3 + 1.68e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-1.64e4 - 1.19e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-9.71e3 + 7.05e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (6.05e3 + 1.86e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-2.39e4 + 7.36e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (432. - 314. i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.75e4 + 5.40e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (1.99e4 + 6.13e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (4.18e4 - 3.03e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (1.14e4 - 3.51e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−11.90983221594667753836282147775, −10.80297276334368542970282279774, −9.832186132984924166318473206251, −8.845649334617321059068597313934, −7.59929509882684317349340590620, −6.58772962740485304704068219796, −6.09555023314356253334227358842, −3.71285414140051094078631149073, −2.43327326641781223167213281947, −0.46344210363509977616739825946,
1.00547353906100471207984689123, 2.97894606284658706371082242097, 4.07749959808567328392796504642, 5.60205001453252305693531187895, 7.06591653636134560269652594673, 8.465484303450441288968870185396, 9.277928539289301476412874372373, 9.937167108374362446556724714982, 11.07164638960983319614258153145, 12.36951058566091858699126701167
