L(s) = 1 | − 0.351i·2-s + (2.28 + 2.28i)3-s + 7.87·4-s + (−9.32 − 9.32i)5-s + (0.800 − 0.800i)6-s + (−23.5 + 23.5i)7-s − 5.57i·8-s − 16.5i·9-s + (−3.27 + 3.27i)10-s + (9.63 − 9.63i)11-s + (17.9 + 17.9i)12-s − 5.12·13-s + (8.26 + 8.26i)14-s − 42.5i·15-s + 61.0·16-s + (44.3 + 54.2i)17-s + ⋯ |
L(s) = 1 | − 0.124i·2-s + (0.439 + 0.439i)3-s + 0.984·4-s + (−0.834 − 0.834i)5-s + (0.0544 − 0.0544i)6-s + (−1.27 + 1.27i)7-s − 0.246i·8-s − 0.614i·9-s + (−0.103 + 0.103i)10-s + (0.264 − 0.264i)11-s + (0.432 + 0.432i)12-s − 0.109·13-s + (0.157 + 0.157i)14-s − 0.732i·15-s + 0.954·16-s + (0.632 + 0.774i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0218i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.14108 + 0.0124600i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14108 + 0.0124600i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-44.3 - 54.2i)T \) |
good | 2 | \( 1 + 0.351iT - 8T^{2} \) |
| 3 | \( 1 + (-2.28 - 2.28i)T + 27iT^{2} \) |
| 5 | \( 1 + (9.32 + 9.32i)T + 125iT^{2} \) |
| 7 | \( 1 + (23.5 - 23.5i)T - 343iT^{2} \) |
| 11 | \( 1 + (-9.63 + 9.63i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + 5.12T + 2.19e3T^{2} \) |
| 19 | \( 1 - 38.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-13.2 + 13.2i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 + (149. + 149. i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 + (-165. - 165. i)T + 2.97e4iT^{2} \) |
| 37 | \( 1 + (-1.96 - 1.96i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + (214. - 214. i)T - 6.89e4iT^{2} \) |
| 43 | \( 1 + 149. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 366.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 499. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 507. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + (23.3 - 23.3i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 - 442.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (336. + 336. i)T + 3.57e5iT^{2} \) |
| 73 | \( 1 + (520. + 520. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + (151. - 151. i)T - 4.93e5iT^{2} \) |
| 83 | \( 1 - 1.18e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 325.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-169. - 169. i)T + 9.12e5iT^{2} \) |
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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
−18.96784474015670650225696139595, −16.65505715293710863374442790654, −15.74297704337318157948957430948, −15.02919414624140641179028646484, −12.55365951775387738372626525003, −11.90381463012337046776509767745, −9.793511341496901751846460700170, −8.419785836937399860036460283773, −6.20715737268211484634574494125, −3.36869997447903686472112275929,
3.21084383272816080884314557330, 6.90363798133140020169176602221, 7.51881575581137225411035006800, 10.19228075879783336397582502162, 11.44915215809469945688305825528, 13.13855181094703508558654024490, 14.52731742111301042910083823201, 15.89675947344895766489601758415, 16.82232628559049666276823238935, 18.94661598236584128982643303434