| L(s) = 1 | + (−0.221 − 1.39i)2-s + (2.68 + 1.36i)3-s + (−1.90 + 0.618i)4-s + (4.84 + 1.24i)5-s + (1.31 − 4.04i)6-s + (−3.67 − 3.67i)7-s + (1.28 + 2.52i)8-s + (0.0358 + 0.0492i)9-s + (0.661 − 7.04i)10-s + (−4.26 − 3.09i)11-s + (−5.94 − 0.941i)12-s + (−1.52 + 9.64i)13-s + (−4.32 + 5.94i)14-s + (11.2 + 9.94i)15-s + (3.23 − 2.35i)16-s + (−19.6 + 10.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.110 − 0.698i)2-s + (0.894 + 0.455i)3-s + (−0.475 + 0.154i)4-s + (0.968 + 0.248i)5-s + (0.219 − 0.674i)6-s + (−0.524 − 0.524i)7-s + (0.160 + 0.315i)8-s + (0.00397 + 0.00547i)9-s + (0.0661 − 0.704i)10-s + (−0.387 − 0.281i)11-s + (−0.495 − 0.0784i)12-s + (−0.117 + 0.742i)13-s + (−0.308 + 0.424i)14-s + (0.753 + 0.663i)15-s + (0.202 − 0.146i)16-s + (−1.15 + 0.589i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.31712 - 0.315428i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31712 - 0.315428i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.221 + 1.39i)T \) |
| 5 | \( 1 + (-4.84 - 1.24i)T \) |
| good | 3 | \( 1 + (-2.68 - 1.36i)T + (5.29 + 7.28i)T^{2} \) |
| 7 | \( 1 + (3.67 + 3.67i)T + 49iT^{2} \) |
| 11 | \( 1 + (4.26 + 3.09i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (1.52 - 9.64i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (19.6 - 10.0i)T + (169. - 233. i)T^{2} \) |
| 19 | \( 1 + (-11.5 - 3.74i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (39.5 - 6.26i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-29.8 + 9.71i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-14.9 + 45.9i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-57.5 - 9.11i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (32.9 - 23.9i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (31.9 - 31.9i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-9.82 + 19.2i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-67.5 - 34.4i)T + (1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (16.7 + 23.0i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-41.1 - 29.8i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-10.9 + 5.59i)T + (2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-30.7 - 94.5i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-3.58 + 0.568i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-76.6 + 24.8i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (32.5 + 63.7i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-32.5 + 44.7i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (45.0 - 88.4i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−14.98623550289011325754773014534, −13.80839792894407516627130663742, −13.34534498547099531300179806816, −11.62713018351658137073110755843, −10.12510176116832773774673328236, −9.582101745849381444587638884744, −8.282180500941148491446017832415, −6.27952179245560580901441655538, −4.06449505979950582610284848831, −2.48563933555529104470691546681,
2.54716488410991714957692759486, 5.23618524843760271724395611082, 6.65580878188749466521571136646, 8.140838915469804010114925811006, 9.115230852552857286761897013624, 10.24233624342708573161528782986, 12.43780790356040121689142834824, 13.48771779354533099714073745972, 14.09586235050030197124078936677, 15.41171176417935303471994618614
