| L(s) = 1 | − 2.36·2-s + 0.429·3-s + 3.58·4-s − 1.05·5-s − 1.01·6-s + 1.83·7-s − 3.75·8-s − 2.81·9-s + 2.49·10-s − 2.02·11-s + 1.53·12-s − 1.26·13-s − 4.33·14-s − 0.453·15-s + 1.69·16-s + 5.49·17-s + 6.65·18-s − 2.74·19-s − 3.79·20-s + 0.786·21-s + 4.78·22-s − 7.47·23-s − 1.61·24-s − 3.88·25-s + 2.99·26-s − 2.49·27-s + 6.57·28-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 0.247·3-s + 1.79·4-s − 0.472·5-s − 0.414·6-s + 0.692·7-s − 1.32·8-s − 0.938·9-s + 0.790·10-s − 0.610·11-s + 0.444·12-s − 0.351·13-s − 1.15·14-s − 0.117·15-s + 0.424·16-s + 1.33·17-s + 1.56·18-s − 0.629·19-s − 0.848·20-s + 0.171·21-s + 1.02·22-s − 1.55·23-s − 0.328·24-s − 0.776·25-s + 0.586·26-s − 0.480·27-s + 1.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4448034355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4448034355\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 3 | \( 1 - 0.429T + 3T^{2} \) |
| 5 | \( 1 + 1.05T + 5T^{2} \) |
| 7 | \( 1 - 1.83T + 7T^{2} \) |
| 11 | \( 1 + 2.02T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 - 5.49T + 17T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 - 5.15T + 29T^{2} \) |
| 31 | \( 1 - 1.37T + 31T^{2} \) |
| 37 | \( 1 - 3.18T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 1.04T + 43T^{2} \) |
| 47 | \( 1 + 9.22T + 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 - 5.42T + 59T^{2} \) |
| 61 | \( 1 - 4.56T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 2.17T + 71T^{2} \) |
| 73 | \( 1 + 8.19T + 73T^{2} \) |
| 79 | \( 1 + 3.60T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
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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
−7.88387606469666558126957391868, −7.53402711738817309314203021144, −6.57497441482313099263173096967, −5.83732986386363231692751084031, −5.06815912907675404027222244245, −4.09625664437234181464362825612, −3.10830221167526356122140314913, −2.33041215504276785474485934040, −1.58763450958012317847775742179, −0.40317525636576498760838785002,
0.40317525636576498760838785002, 1.58763450958012317847775742179, 2.33041215504276785474485934040, 3.10830221167526356122140314913, 4.09625664437234181464362825612, 5.06815912907675404027222244245, 5.83732986386363231692751084031, 6.57497441482313099263173096967, 7.53402711738817309314203021144, 7.88387606469666558126957391868
