| L(s) = 1 | + (0.833 − 3.65i)2-s + (1.32 − 1.65i)3-s + (−5.44 − 2.62i)4-s + (6.56 − 8.23i)5-s + (−4.95 − 6.21i)6-s + (−18.3 + 2.83i)7-s + (4.56 − 5.72i)8-s + (5.00 + 21.9i)9-s + (−24.6 − 30.8i)10-s + (5.52 − 24.1i)11-s + (−11.5 + 5.56i)12-s + (−15.5 + 68.1i)13-s + (−4.90 + 69.2i)14-s + (−4.97 − 21.7i)15-s + (−47.2 − 59.2i)16-s + (67.9 − 32.7i)17-s + ⋯ |
| L(s) = 1 | + (0.294 − 1.29i)2-s + (0.254 − 0.319i)3-s + (−0.680 − 0.327i)4-s + (0.587 − 0.736i)5-s + (−0.337 − 0.422i)6-s + (−0.988 + 0.153i)7-s + (0.201 − 0.253i)8-s + (0.185 + 0.812i)9-s + (−0.777 − 0.975i)10-s + (0.151 − 0.663i)11-s + (−0.277 + 0.133i)12-s + (−0.331 + 1.45i)13-s + (−0.0936 + 1.32i)14-s + (−0.0855 − 0.374i)15-s + (−0.738 − 0.926i)16-s + (0.970 − 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.847174 - 1.52781i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.847174 - 1.52781i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (18.3 - 2.83i)T \) |
| good | 2 | \( 1 + (-0.833 + 3.65i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (-1.32 + 1.65i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-6.56 + 8.23i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (-5.52 + 24.1i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (15.5 - 68.1i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (-67.9 + 32.7i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 - 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-22.2 - 10.6i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (144. - 69.6i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 + 43.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + (181. - 87.3i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (105. - 132. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (245. + 307. i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (53.8 - 235. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (377. + 181. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (-92.9 - 116. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (-164. + 79.0i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 + 968.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-328. - 158. i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-109. - 478. i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (109. + 480. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (190. + 836. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + 552.T + 9.12e5T^{2} \) |
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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
−13.96573580874179333902482320296, −13.43752065905637231635772524085, −12.41829855678643424200499607659, −11.44660697932802530353598764662, −9.927827238773881275310646800097, −9.150455759431597782388845375516, −7.15789646715506835351509743148, −5.17989795018576927940432765948, −3.25843099527220218449679203717, −1.56765969814381662436794749991,
3.33521870413544123588829781808, 5.49994073199561517540861469331, 6.61342922844510983820945958624, 7.67414949796945430338438321953, 9.522006662173778421891429174714, 10.35610957635454749634853253090, 12.40053461687206152802055518029, 13.63112610525349166279294385454, 14.74070410580521817933167435140, 15.28300061021758590568808392174
