| L(s) = 1 | + (0.347 − 1.96i)2-s + (−2.29 − 1.92i)3-s + (−3.75 − 1.36i)4-s + (−9.75 + 3.55i)5-s + (−4.59 + 3.85i)6-s + (17.5 + 30.3i)7-s + (−4 + 6.92i)8-s + (1.56 + 8.86i)9-s + (3.60 + 20.4i)10-s + (−0.139 + 0.241i)11-s + (6 + 10.3i)12-s + (57.0 − 47.8i)13-s + (65.8 − 23.9i)14-s + (29.2 + 10.6i)15-s + (12.2 + 10.2i)16-s + (−13.7 + 77.7i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.442 − 0.371i)3-s + (−0.469 − 0.171i)4-s + (−0.872 + 0.317i)5-s + (−0.312 + 0.262i)6-s + (0.945 + 1.63i)7-s + (−0.176 + 0.306i)8-s + (0.0578 + 0.328i)9-s + (0.114 + 0.646i)10-s + (−0.00382 + 0.00662i)11-s + (0.144 + 0.249i)12-s + (1.21 − 1.02i)13-s + (1.25 − 0.457i)14-s + (0.503 + 0.183i)15-s + (0.191 + 0.160i)16-s + (−0.195 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.05367 + 0.311631i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05367 + 0.311631i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (48.8 - 66.8i)T \) |
| good | 5 | \( 1 + (9.75 - 3.55i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-17.5 - 30.3i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (0.139 - 0.241i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-57.0 + 47.8i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (13.7 - 77.7i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 23 | \( 1 + (42.4 + 15.4i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-45.8 - 259. i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-153. - 266. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 91.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + (270. + 227. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-153. + 55.8i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (17.4 + 98.6i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + (319. + 116. i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-118. + 671. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (239. + 87.3i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (19.6 + 111. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-911. + 331. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + (-32.7 - 27.4i)T + (6.75e4 + 3.83e5i)T^{2} \) |
| 79 | \( 1 + (458. + 384. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-186. - 322. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (488. - 410. i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (36.7 - 208. i)T + (-8.57e5 - 3.12e5i)T^{2} \) |
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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
−12.65553325880674141844725357442, −12.22437975930790991940784664096, −11.18902772914477879755217165495, −10.55599187280517014603877880857, −8.585675548951213103986231188890, −8.160581961582204968746416938539, −6.20722084442233639631951537716, −5.12045584386526097101936400091, −3.44020174990563120774524922140, −1.70233812367655571970082627768,
0.65150023069683218610596574602, 4.21381436810797875619225755661, 4.44790683466095572661054907063, 6.35605628982832809345372170504, 7.50279070285162647055461328665, 8.383483676493237486236857818082, 9.802814426487842447577053108635, 11.23114240534563429422337589656, 11.60686750166843832684533056233, 13.42690504668234317409046633536
