| L(s) = 1 | + (−3.23 + 2.35i)2-s + (−20.9 − 15.2i)3-s + (4.94 − 15.2i)4-s + 78.8·5-s + 103.·6-s + (−0.961 + 2.96i)7-s + (19.7 + 60.8i)8-s + (132. + 406. i)9-s + (−255. + 185. i)10-s + (178. − 549. i)11-s + (−335. + 243. i)12-s + (−373. − 271. i)13-s + (−3.84 − 11.8i)14-s + (−1.65e3 − 1.20e3i)15-s + (−207. − 150. i)16-s + (−62.9 − 193. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−1.34 − 0.976i)3-s + (0.154 − 0.475i)4-s + 1.41·5-s + 1.17·6-s + (−0.00741 + 0.0228i)7-s + (0.109 + 0.336i)8-s + (0.544 + 1.67i)9-s + (−0.806 + 0.586i)10-s + (0.444 − 1.36i)11-s + (−0.672 + 0.488i)12-s + (−0.612 − 0.445i)13-s + (−0.00524 − 0.0161i)14-s + (−1.89 − 1.37i)15-s + (−0.202 − 0.146i)16-s + (−0.0528 − 0.162i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.218244 - 0.577256i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.218244 - 0.577256i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 - 2.35i)T \) |
| 31 | \( 1 + (2.05e3 + 4.94e3i)T \) |
| good | 3 | \( 1 + (20.9 + 15.2i)T + (75.0 + 231. i)T^{2} \) |
| 5 | \( 1 - 78.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + (0.961 - 2.96i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 11 | \( 1 + (-178. + 549. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (373. + 271. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (62.9 + 193. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (997. - 724. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (782. + 2.40e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (6.58e3 - 4.78e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 37 | \( 1 + 1.04e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-7.27e3 + 5.28e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + (-1.10e4 + 7.99e3i)T + (4.54e7 - 1.39e8i)T^{2} \) |
| 47 | \( 1 + (1.76e4 + 1.28e4i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (3.55e3 + 1.09e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-2.31e4 - 1.68e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 - 1.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.78e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-7.68e3 - 2.36e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-1.32e4 + 4.07e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (3.21e4 + 9.90e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (8.12e4 - 5.90e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (1.31e4 - 4.05e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-3.14e4 + 9.68e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−13.47552479358049939829974033317, −12.48608538180816440588698412217, −11.14996472564416473453969317284, −10.26098710658785470545935243880, −8.809451457607576613245497608564, −7.18293112737287644825708935310, −6.04320183489147958886594590561, −5.53135885852877708383101820331, −1.87467607289005799910306301620, −0.40061140441106649812189289832,
1.84195730931488770662709412057, 4.38145107954036494073094048886, 5.67193701606571549351464440246, 6.93839246038990708557570199586, 9.423663572557875201548799987112, 9.773246653114667098872159038419, 10.81277527615751548993292308851, 11.89749657669204034141116674653, 12.97734739088520850916122301225, 14.60039484672728177462551624793
