| L(s) = 1 | + (2.15 + 3.72i)2-s + (−4.60 − 2.40i)3-s + (−5.26 + 9.12i)4-s − 15.9·5-s + (−0.955 − 22.3i)6-s + (−16.8 + 7.61i)7-s − 10.9·8-s + (15.4 + 22.1i)9-s + (−34.4 − 59.6i)10-s − 12.3·11-s + (46.2 − 29.3i)12-s + (35.8 + 62.0i)13-s + (−64.7 − 46.5i)14-s + (73.6 + 38.4i)15-s + (18.6 + 32.2i)16-s + (−42.2 − 73.1i)17-s + ⋯ |
| L(s) = 1 | + (0.761 + 1.31i)2-s + (−0.886 − 0.462i)3-s + (−0.658 + 1.14i)4-s − 1.43·5-s + (−0.0649 − 1.52i)6-s + (−0.911 + 0.411i)7-s − 0.483·8-s + (0.572 + 0.820i)9-s + (−1.08 − 1.88i)10-s − 0.338·11-s + (1.11 − 0.706i)12-s + (0.764 + 1.32i)13-s + (−1.23 − 0.888i)14-s + (1.26 + 0.661i)15-s + (0.290 + 0.503i)16-s + (−0.602 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.124458 - 0.617096i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.124458 - 0.617096i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.60 + 2.40i)T \) |
| 7 | \( 1 + (16.8 - 7.61i)T \) |
| good | 2 | \( 1 + (-2.15 - 3.72i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + 15.9T + 125T^{2} \) |
| 11 | \( 1 + 12.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-35.8 - 62.0i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (42.2 + 73.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-13.6 + 23.6i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 40.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + (99.2 - 171. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (146. - 253. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-58.7 + 101. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (18.6 + 32.2i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-122. + 212. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (45.8 + 79.3i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-85.8 - 148. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-51.2 + 88.8i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-290. - 503. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-51.5 + 89.3i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 204.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-580. - 1.00e3i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (310. + 537. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-67.9 + 117. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (710. - 1.23e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (559. - 968. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−15.47609637107428827171135875906, −14.02525076317182715162355013719, −12.96286467706998202731991803714, −12.01595621370601793083275033420, −10.98549615442986356501298000856, −8.789558565585842007149286369448, −7.28150467122497059438296932823, −6.73446353124758621501842022005, −5.31344670102777579430112463081, −3.97434343255291625200649544655,
0.37756391882832123065985807691, 3.49347340761853025871329759083, 4.20349013593676284449894433255, 5.91502860553969646001577008847, 7.80311950765059054475264935227, 9.885290391666823215776483994934, 10.84398224998063775095744148835, 11.48165394968845186352528661773, 12.65198354084289905901237616962, 13.14758145400398236496068214634
