L(s) = 1 | + (−0.161 − 0.161i)2-s + (3.15 + 7.61i)3-s − 7.94i·4-s + (2.54 − 1.05i)5-s + (0.718 − 1.73i)6-s + (−19.8 − 8.23i)7-s + (−2.56 + 2.56i)8-s + (−28.9 + 28.9i)9-s + (−0.579 − 0.239i)10-s + (20.8 − 50.3i)11-s + (60.5 − 25.0i)12-s + 52.4i·13-s + (1.87 + 4.53i)14-s + (16.0 + 16.0i)15-s − 62.7·16-s + (33.5 + 61.5i)17-s + ⋯ |
L(s) = 1 | + (−0.0569 − 0.0569i)2-s + (0.606 + 1.46i)3-s − 0.993i·4-s + (0.227 − 0.0941i)5-s + (0.0488 − 0.118i)6-s + (−1.07 − 0.444i)7-s + (−0.113 + 0.113i)8-s + (−1.07 + 1.07i)9-s + (−0.0183 − 0.00758i)10-s + (0.572 − 1.38i)11-s + (1.45 − 0.602i)12-s + 1.11i·13-s + (0.0358 + 0.0864i)14-s + (0.275 + 0.275i)15-s − 0.980·16-s + (0.478 + 0.878i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.08914 + 0.233831i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08914 + 0.233831i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-33.5 - 61.5i)T \) |
good | 2 | \( 1 + (0.161 + 0.161i)T + 8iT^{2} \) |
| 3 | \( 1 + (-3.15 - 7.61i)T + (-19.0 + 19.0i)T^{2} \) |
| 5 | \( 1 + (-2.54 + 1.05i)T + (88.3 - 88.3i)T^{2} \) |
| 7 | \( 1 + (19.8 + 8.23i)T + (242. + 242. i)T^{2} \) |
| 11 | \( 1 + (-20.8 + 50.3i)T + (-941. - 941. i)T^{2} \) |
| 13 | \( 1 - 52.4iT - 2.19e3T^{2} \) |
| 19 | \( 1 + (-13.8 - 13.8i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 + (5.33 - 12.8i)T + (-8.60e3 - 8.60e3i)T^{2} \) |
| 29 | \( 1 + (-64.6 + 26.7i)T + (1.72e4 - 1.72e4i)T^{2} \) |
| 31 | \( 1 + (63.9 + 154. i)T + (-2.10e4 + 2.10e4i)T^{2} \) |
| 37 | \( 1 + (76.0 + 183. i)T + (-3.58e4 + 3.58e4i)T^{2} \) |
| 41 | \( 1 + (-401. - 166. i)T + (4.87e4 + 4.87e4i)T^{2} \) |
| 43 | \( 1 + (-89.9 + 89.9i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + 207. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (220. + 220. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (407. - 407. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (-72.4 - 29.9i)T + (1.60e5 + 1.60e5i)T^{2} \) |
| 67 | \( 1 - 359.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-81.5 - 196. i)T + (-2.53e5 + 2.53e5i)T^{2} \) |
| 73 | \( 1 + (-26.8 + 11.1i)T + (2.75e5 - 2.75e5i)T^{2} \) |
| 79 | \( 1 + (-327. + 790. i)T + (-3.48e5 - 3.48e5i)T^{2} \) |
| 83 | \( 1 + (-9.67 - 9.67i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 651. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (1.10e3 - 458. i)T + (6.45e5 - 6.45e5i)T^{2} \) |
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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
−19.11953457980084304618984746641, −16.71533390350133679044071498810, −15.94446122965336547875870431583, −14.60012879705141850551372015477, −13.69616040899096529874629439079, −11.07807009325617276065682987141, −9.857099337988617231862647489296, −9.061976897313907715867345366026, −5.99428858372868210881898298167, −3.83434324346232986494305054609,
2.80811572346532257613209573940, 6.69215851517051371857735440021, 7.80911293890994938460299409290, 9.411861261245147617561087722200, 12.30467969059534034536152254567, 12.66865075572563584507152977112, 14.01395537089286119703596445981, 15.77742404286221559423046356837, 17.51066461239848595441339818998, 18.21839307991142742185535940248