| L(s) = 1 | + (−0.347 + 1.96i)2-s + (2.29 + 1.92i)3-s + (−3.75 − 1.36i)4-s + (−1.70 + 0.618i)5-s + (−4.59 + 3.85i)6-s + (8.26 + 14.3i)7-s + (4 − 6.92i)8-s + (1.56 + 8.86i)9-s + (−0.628 − 3.56i)10-s + (−21.4 + 37.1i)11-s + (−6 − 10.3i)12-s + (−53.5 + 44.9i)13-s + (−31.0 + 11.3i)14-s + (−5.10 − 1.85i)15-s + (12.2 + 10.2i)16-s + (11.6 − 65.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.442 + 0.371i)3-s + (−0.469 − 0.171i)4-s + (−0.152 + 0.0553i)5-s + (−0.312 + 0.262i)6-s + (0.446 + 0.773i)7-s + (0.176 − 0.306i)8-s + (0.0578 + 0.328i)9-s + (−0.0198 − 0.112i)10-s + (−0.587 + 1.01i)11-s + (−0.144 − 0.249i)12-s + (−1.14 + 0.959i)13-s + (−0.593 + 0.215i)14-s + (−0.0878 − 0.0319i)15-s + (0.191 + 0.160i)16-s + (0.166 − 0.941i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.547i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.384949 + 1.29230i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.384949 + 1.29230i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.347 - 1.96i)T \) |
| 3 | \( 1 + (-2.29 - 1.92i)T \) |
| 19 | \( 1 + (-80.9 - 17.6i)T \) |
| good | 5 | \( 1 + (1.70 - 0.618i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-8.26 - 14.3i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (21.4 - 37.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (53.5 - 44.9i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-11.6 + 65.9i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 23 | \( 1 + (91.5 + 33.3i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (7.92 + 44.9i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-91.8 - 159. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 313.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-24.0 - 20.1i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-442. + 161. i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-52.3 - 296. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + (-85.6 - 31.1i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-111. + 632. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-224. - 81.7i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-80.3 - 455. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-596. + 217. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + (-524. - 439. i)T + (6.75e4 + 3.83e5i)T^{2} \) |
| 79 | \( 1 + (746. + 626. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (247. + 428. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (898. - 754. i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (71.4 - 405. i)T + (-8.57e5 - 3.12e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.96207741775272629493621556183, −12.48660818299778681621508429547, −11.56966506437882085303961604771, −9.890432735443669571044641973270, −9.334657292875251846466239810515, −7.964252068901599408430748652752, −7.17121694486295790034621503398, −5.42974861949890726818424820199, −4.44052696286985410304374907224, −2.38270788589224854461771388380,
0.73029279782346207939437301682, 2.63131150172492737345940459956, 4.04882700179075984382829280588, 5.67805778003913359800225410921, 7.65874938190337079849790311611, 8.115806334810304069331356441444, 9.671821192016294403528227228807, 10.57359728164091223142300190343, 11.64385463909435720741536698975, 12.71049020718470725725499211701
