| L(s) = 1 | + (−1.27 + 1.18i)2-s + (18.3 − 2.76i)3-s + (−2.16 + 28.8i)4-s + (−17.7 + 45.3i)5-s + (−20.0 + 25.1i)6-s + (−121. + 44.3i)7-s + (−66.0 − 82.8i)8-s + (96.3 − 29.7i)9-s + (−30.9 − 78.8i)10-s + (172. + 53.2i)11-s + (40.1 + 535. i)12-s + (−0.626 − 2.74i)13-s + (102. − 200. i)14-s + (−200. + 880. i)15-s + (−734. − 110. i)16-s + (−314. + 214. i)17-s + ⋯ |
| L(s) = 1 | + (−0.225 + 0.208i)2-s + (1.17 − 0.177i)3-s + (−0.0676 + 0.903i)4-s + (−0.318 + 0.811i)5-s + (−0.227 + 0.285i)6-s + (−0.939 + 0.341i)7-s + (−0.364 − 0.457i)8-s + (0.396 − 0.122i)9-s + (−0.0977 − 0.249i)10-s + (0.430 + 0.132i)11-s + (0.0805 + 1.07i)12-s + (−0.00102 − 0.00450i)13-s + (0.140 − 0.273i)14-s + (−0.230 + 1.01i)15-s + (−0.717 − 0.108i)16-s + (−0.264 + 0.180i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.790240 + 1.35687i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.790240 + 1.35687i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (121. - 44.3i)T \) |
| good | 2 | \( 1 + (1.27 - 1.18i)T + (2.39 - 31.9i)T^{2} \) |
| 3 | \( 1 + (-18.3 + 2.76i)T + (232. - 71.6i)T^{2} \) |
| 5 | \( 1 + (17.7 - 45.3i)T + (-2.29e3 - 2.12e3i)T^{2} \) |
| 11 | \( 1 + (-172. - 53.2i)T + (1.33e5 + 9.07e4i)T^{2} \) |
| 13 | \( 1 + (0.626 + 2.74i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (314. - 214. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (-999. - 1.73e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-1.11e3 - 760. i)T + (2.35e6 + 5.99e6i)T^{2} \) |
| 29 | \( 1 + (-6.88e3 - 3.31e3i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (-5.20e3 + 9.00e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-9.82 - 131. i)T + (-6.85e7 + 1.03e7i)T^{2} \) |
| 41 | \( 1 + (3.62e3 + 4.54e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (2.97e3 - 3.73e3i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (5.99e3 - 5.56e3i)T + (1.71e7 - 2.28e8i)T^{2} \) |
| 53 | \( 1 + (-607. + 8.11e3i)T + (-4.13e8 - 6.23e7i)T^{2} \) |
| 59 | \( 1 + (-8.96e3 - 2.28e4i)T + (-5.24e8 + 4.86e8i)T^{2} \) |
| 61 | \( 1 + (2.86e3 + 3.82e4i)T + (-8.35e8 + 1.25e8i)T^{2} \) |
| 67 | \( 1 + (-3.00e3 + 5.20e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (3.81e4 - 1.83e4i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-5.74e4 - 5.32e4i)T + (1.54e8 + 2.06e9i)T^{2} \) |
| 79 | \( 1 + (3.19e4 + 5.53e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (1.45e4 - 6.39e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (-5.84e4 + 1.80e4i)T + (4.61e9 - 3.14e9i)T^{2} \) |
| 97 | \( 1 - 1.09e5T + 8.58e9T^{2} \) |
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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
−14.98465996782284537953192573001, −13.88753550397157410711280231642, −12.82880364261361496274976148069, −11.66990550152395770951660923468, −9.829735511569934988091021445468, −8.734562331852315434066627973744, −7.68527018073496438358477429077, −6.56601871730805061242563632334, −3.64150626378470875484532243664, −2.76962663549645560694157560031,
0.77716320033719820379983859556, 2.91564520719032779900577110489, 4.69861460021288108180476311072, 6.63582136140643033514840021950, 8.537978299182607282674756736368, 9.233604168222908841382933886021, 10.28140150732084055037175525737, 11.88710184272927819334550050464, 13.39225953084386555318503442061, 14.12513771025657350404012207099
