| L(s) = 1 | + (1.46e11 − 2.53e11i)2-s + (1.40e17 + 7.67e17i)3-s + (−2.37e22 − 4.12e22i)4-s + (−1.57e26 − 2.72e26i)5-s + (2.14e29 + 7.64e28i)6-s + (−3.92e31 + 6.79e31i)7-s + (−2.86e33 + 5.76e17i)8-s + (−5.68e35 + 2.16e35i)9-s + (−9.19e37 + 4.72e21i)10-s + (6.51e38 − 1.12e39i)11-s + (2.82e40 − 2.40e40i)12-s + (−1.83e41 − 3.18e41i)13-s + (1.14e43 + 1.98e43i)14-s + (1.86e44 − 1.59e44i)15-s + (4.80e44 − 8.31e44i)16-s + 9.97e44·17-s + ⋯ |
| L(s) = 1 | + (0.751 − 1.30i)2-s + (0.180 + 0.983i)3-s + (−0.629 − 1.09i)4-s + (−0.967 − 1.67i)5-s + (1.41 + 0.503i)6-s + (−0.798 + 1.38i)7-s − 0.390·8-s + (−0.934 + 0.355i)9-s − 2.90·10-s + (0.577 − 0.999i)11-s + (0.959 − 0.816i)12-s + (−0.309 − 0.536i)13-s + (1.20 + 2.08i)14-s + (1.47 − 1.25i)15-s + (0.336 − 0.582i)16-s + 0.0719·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(76-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+75/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(38)\) |
\(\approx\) |
\(1.905944681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.905944681\) |
| \(L(\frac{77}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.40e17 - 7.67e17i)T \) |
| good | 2 | \( 1 + (-1.46e11 + 2.53e11i)T + (-1.88e22 - 3.27e22i)T^{2} \) |
| 5 | \( 1 + (1.57e26 + 2.72e26i)T + (-1.32e52 + 2.29e52i)T^{2} \) |
| 7 | \( 1 + (3.92e31 - 6.79e31i)T + (-1.20e63 - 2.08e63i)T^{2} \) |
| 11 | \( 1 + (-6.51e38 + 1.12e39i)T + (-6.35e77 - 1.10e78i)T^{2} \) |
| 13 | \( 1 + (1.83e41 + 3.18e41i)T + (-1.75e83 + 3.04e83i)T^{2} \) |
| 17 | \( 1 - 9.97e44T + 1.92e92T^{2} \) |
| 19 | \( 1 - 1.26e48T + 8.06e95T^{2} \) |
| 23 | \( 1 + (-9.82e49 - 1.70e50i)T + (-6.73e101 + 1.16e102i)T^{2} \) |
| 29 | \( 1 + (2.79e54 - 4.84e54i)T + (-2.39e109 - 4.14e109i)T^{2} \) |
| 31 | \( 1 + (1.17e55 + 2.03e55i)T + (-3.55e111 + 6.16e111i)T^{2} \) |
| 37 | \( 1 + (-6.62e58 - 2.69e4i)T + 4.12e117T^{2} \) |
| 41 | \( 1 + (-9.29e59 - 1.60e60i)T + (-4.54e120 + 7.87e120i)T^{2} \) |
| 43 | \( 1 + (-9.49e60 + 1.64e61i)T + (-1.61e122 - 2.80e122i)T^{2} \) |
| 47 | \( 1 + (-2.49e62 + 4.32e62i)T + (-1.27e125 - 2.21e125i)T^{2} \) |
| 53 | \( 1 + (-6.32e64 + 1.78e10i)T + 2.09e129T^{2} \) |
| 59 | \( 1 + (4.88e65 + 8.46e65i)T + (-3.25e132 + 5.64e132i)T^{2} \) |
| 61 | \( 1 + (-5.02e66 + 8.70e66i)T + (-3.96e133 - 6.87e133i)T^{2} \) |
| 67 | \( 1 + (1.59e67 + 2.76e67i)T + (-4.51e136 + 7.81e136i)T^{2} \) |
| 71 | \( 1 + (-4.41e69 + 9.00e13i)T + 6.98e138T^{2} \) |
| 73 | \( 1 + (-2.97e69 - 2.99e15i)T + 5.61e139T^{2} \) |
| 79 | \( 1 + (-6.13e70 + 1.06e71i)T + (-1.04e142 - 1.81e142i)T^{2} \) |
| 83 | \( 1 + (7.76e71 - 1.34e72i)T + (-4.26e143 - 7.38e143i)T^{2} \) |
| 89 | \( 1 + (-3.63e71 - 5.23e17i)T + 1.60e146T^{2} \) |
| 97 | \( 1 + (1.36e74 - 2.36e74i)T + (-5.09e148 - 8.81e148i)T^{2} \) |
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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
−9.589966091794683310882499232779, −9.078822514167352603987714744209, −8.049506991844070047162700822845, −5.44897123376613836425895626141, −5.28226721711125886076609589962, −3.95067420029180253880916735387, −3.43319295941646982073911011933, −2.51181778124574105508093504863, −1.06314826253951143718946918198, −0.28501953424843549871644479071,
0.976995766811358981327437165822, 2.57925477111245575574763388924, 3.63782786046527377890357437530, 4.24566708274380729091053320526, 6.05292300307392772793435266703, 6.92954787897505064635490925200, 7.21840128139607071836094864417, 7.76830829013790568442480620426, 9.784535224485878169177680772875, 11.11201072044087607111054873244
