| L(s) = 1 | + (−2.47 − 0.141i)2-s + (−1.86 + 5.75i)3-s + (−1.85 − 0.212i)4-s + (5.25 + 9.96i)5-s + (5.43 − 13.9i)6-s + (20.5 + 8.69i)7-s + (24.0 + 4.16i)8-s + (−7.75 − 5.63i)9-s + (−11.5 − 25.3i)10-s + (−13.7 + 33.8i)11-s + (4.68 − 10.2i)12-s + (−14.8 + 8.40i)13-s + (−49.6 − 24.4i)14-s + (−67.1 + 11.6i)15-s + (−44.4 − 10.3i)16-s + (−21.3 + 7.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.874 − 0.0499i)2-s + (−0.359 + 1.10i)3-s + (−0.231 − 0.0265i)4-s + (0.470 + 0.891i)5-s + (0.369 − 0.949i)6-s + (1.11 + 0.469i)7-s + (1.06 + 0.184i)8-s + (−0.287 − 0.208i)9-s + (−0.366 − 0.803i)10-s + (−0.376 + 0.926i)11-s + (0.112 − 0.246i)12-s + (−0.317 + 0.179i)13-s + (−0.947 − 0.465i)14-s + (−1.15 + 0.200i)15-s + (−0.693 − 0.161i)16-s + (−0.304 + 0.108i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.133741 + 0.790902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.133741 + 0.790902i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (13.7 - 33.8i)T \) |
| good | 2 | \( 1 + (2.47 + 0.141i)T + (7.94 + 0.911i)T^{2} \) |
| 3 | \( 1 + (1.86 - 5.75i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (-5.25 - 9.96i)T + (-70.5 + 103. i)T^{2} \) |
| 7 | \( 1 + (-20.5 - 8.69i)T + (239. + 245. i)T^{2} \) |
| 13 | \( 1 + (14.8 - 8.40i)T + (1.13e3 - 1.88e3i)T^{2} \) |
| 17 | \( 1 + (21.3 - 7.60i)T + (3.80e3 - 3.10e3i)T^{2} \) |
| 19 | \( 1 + (-0.515 + 18.0i)T + (-6.84e3 - 391. i)T^{2} \) |
| 23 | \( 1 + (63.4 + 73.2i)T + (-1.73e3 + 1.20e4i)T^{2} \) |
| 29 | \( 1 + (-108. - 159. i)T + (-8.84e3 + 2.27e4i)T^{2} \) |
| 31 | \( 1 + (-10.3 + 51.2i)T + (-2.74e4 - 1.15e4i)T^{2} \) |
| 37 | \( 1 + (127. - 116. i)T + (4.33e3 - 5.04e4i)T^{2} \) |
| 41 | \( 1 + (-18.9 - 72.1i)T + (-6.00e4 + 3.38e4i)T^{2} \) |
| 43 | \( 1 + (33.8 + 235. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + (-416. + 340. i)T + (2.06e4 - 1.01e5i)T^{2} \) |
| 53 | \( 1 + (-220. + 51.2i)T + (1.33e5 - 6.56e4i)T^{2} \) |
| 59 | \( 1 + (19.2 - 73.4i)T + (-1.78e5 - 1.00e5i)T^{2} \) |
| 61 | \( 1 + (481. - 27.5i)T + (2.25e5 - 2.58e4i)T^{2} \) |
| 67 | \( 1 + (440. - 283. i)T + (1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-200. + 260. i)T + (-9.09e4 - 3.46e5i)T^{2} \) |
| 73 | \( 1 + (-237. + 393. i)T + (-1.81e5 - 3.44e5i)T^{2} \) |
| 79 | \( 1 + (330. - 340. i)T + (-1.40e4 - 4.92e5i)T^{2} \) |
| 83 | \( 1 + (115. - 1.34e3i)T + (-5.63e5 - 9.75e4i)T^{2} \) |
| 89 | \( 1 + (-1.10e3 - 323. i)T + (5.93e5 + 3.81e5i)T^{2} \) |
| 97 | \( 1 + (800. - 1.51e3i)T + (-5.15e5 - 7.53e5i)T^{2} \) |
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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
−13.69357047306675089668637028886, −12.06890569042164963633553595972, −10.70220082592744585686078779358, −10.42565176083119775227336037583, −9.434809639493374948779145825920, −8.357156877766139742442989572306, −7.06674585746163759512746207247, −5.24391672143741651408224916835, −4.41106234431950351847046156522, −2.10553824764434469311630693320,
0.63601798466239895571470259492, 1.62584354772462967176438533968, 4.54992635690107351513602055830, 5.80704540416711543556876147618, 7.44628524720827683185893052074, 8.109066032934253044922704527764, 9.097161014447511991329481332974, 10.35006396655052348020095176758, 11.46772108045305543660219918839, 12.61118108103823651466775737533
