L(s) = 1 | + (−1.11 − 1.11i)2-s + (2.19 + 2.19i)3-s − 1.52i·4-s + (−1.87 − 1.87i)5-s − 4.88i·6-s − 5.07i·7-s + (−6.14 + 6.14i)8-s + 0.630i·9-s + 4.17i·10-s + (−9.24 − 9.24i)11-s + (3.33 − 3.33i)12-s + (2.79 + 2.79i)13-s + (−5.64 + 5.64i)14-s − 8.22i·15-s + 7.60·16-s + (−12.5 + 12.5i)17-s + ⋯ |
L(s) = 1 | + (−0.556 − 0.556i)2-s + (0.731 + 0.731i)3-s − 0.380i·4-s + (−0.374 − 0.374i)5-s − 0.814i·6-s − 0.724i·7-s + (−0.768 + 0.768i)8-s + 0.0701i·9-s + 0.417i·10-s + (−0.840 − 0.840i)11-s + (0.277 − 0.277i)12-s + (0.214 + 0.214i)13-s + (−0.403 + 0.403i)14-s − 0.548i·15-s + 0.475·16-s + (−0.736 + 0.736i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.400594 - 0.882946i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.400594 - 0.882946i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 197 | \( 1 + (6.06 + 196. i)T \) |
good | 2 | \( 1 + (1.11 + 1.11i)T + 4iT^{2} \) |
| 3 | \( 1 + (-2.19 - 2.19i)T + 9iT^{2} \) |
| 5 | \( 1 + (1.87 + 1.87i)T + 25iT^{2} \) |
| 7 | \( 1 + 5.07iT - 49T^{2} \) |
| 11 | \( 1 + (9.24 + 9.24i)T + 121iT^{2} \) |
| 13 | \( 1 + (-2.79 - 2.79i)T + 169iT^{2} \) |
| 17 | \( 1 + (12.5 - 12.5i)T - 289iT^{2} \) |
| 19 | \( 1 + 33.2iT - 361T^{2} \) |
| 23 | \( 1 - 30.3T + 529T^{2} \) |
| 29 | \( 1 + 36.4T + 841T^{2} \) |
| 31 | \( 1 + (5.16 + 5.16i)T + 961iT^{2} \) |
| 37 | \( 1 + 42.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 0.230iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 70.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 55.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 74.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 75.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 83.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + (16.2 + 16.2i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + (65.9 - 65.9i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (-53.0 - 53.0i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + (-39.8 + 39.8i)T - 6.24e3iT^{2} \) |
| 83 | \( 1 - 121. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (81.1 - 81.1i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + 168. iT - 9.40e3T^{2} \) |
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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
−11.39377607880859985183057283987, −10.84403394388024860374109907101, −9.939666037255036639538501247063, −8.869271111878537572164542681326, −8.489156494175677675596142870333, −6.85496984061652156723382527871, −5.25196946762335029852093061605, −3.99206395103038069077303644419, −2.66169053047255502330082588743, −0.59817383931822810092612788912,
2.24272951333048672610759400019, 3.47494695542059710337963504691, 5.42952585662251116690812500243, 7.07376441517262045430074110141, 7.46855231660282221457369602358, 8.449930388144303976735236468298, 9.187533702111006800091228675610, 10.52817686789098658184311763606, 11.86585402531543586525079782322, 12.73226401051962115661531045156