| L(s) = 1 | − 2.68·2-s + 1.49·3-s + 5.23·4-s − 2.86·5-s − 4.03·6-s + 3.78·7-s − 8.69·8-s − 0.752·9-s + 7.70·10-s − 11-s + 7.84·12-s + 1.85·13-s − 10.1·14-s − 4.29·15-s + 12.9·16-s + 3.20·17-s + 2.02·18-s − 4.08·19-s − 14.9·20-s + 5.67·21-s + 2.68·22-s − 2.33·23-s − 13.0·24-s + 3.21·25-s − 4.99·26-s − 5.62·27-s + 19.7·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 0.865·3-s + 2.61·4-s − 1.28·5-s − 1.64·6-s + 1.43·7-s − 3.07·8-s − 0.250·9-s + 2.43·10-s − 0.301·11-s + 2.26·12-s + 0.514·13-s − 2.71·14-s − 1.10·15-s + 3.22·16-s + 0.777·17-s + 0.477·18-s − 0.937·19-s − 3.35·20-s + 1.23·21-s + 0.573·22-s − 0.486·23-s − 2.65·24-s + 0.643·25-s − 0.978·26-s − 1.08·27-s + 3.74·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8245072471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8245072471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 547 | \( 1 - T \) |
| good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 3 | \( 1 - 1.49T + 3T^{2} \) |
| 5 | \( 1 + 2.86T + 5T^{2} \) |
| 7 | \( 1 - 3.78T + 7T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 - 3.20T + 17T^{2} \) |
| 19 | \( 1 + 4.08T + 19T^{2} \) |
| 23 | \( 1 + 2.33T + 23T^{2} \) |
| 29 | \( 1 - 0.511T + 29T^{2} \) |
| 31 | \( 1 + 4.10T + 31T^{2} \) |
| 37 | \( 1 - 6.63T + 37T^{2} \) |
| 41 | \( 1 - 3.51T + 41T^{2} \) |
| 43 | \( 1 - 0.148T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 2.13T + 61T^{2} \) |
| 67 | \( 1 - 0.293T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 0.0439T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 4.14T + 89T^{2} \) |
| 97 | \( 1 - 19.2T + 97T^{2} \) |
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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
−8.261635632931177593859400103507, −7.82838952228452500673863211994, −7.27315734999518814405234028609, −6.30298969327372768593273444330, −5.35684864602253773475513103494, −4.14991867284145974631861688589, −3.37590926823296063185264661604, −2.41781946040259152931033508739, −1.70758030304981534117353216218, −0.60807821632709590429117826353,
0.60807821632709590429117826353, 1.70758030304981534117353216218, 2.41781946040259152931033508739, 3.37590926823296063185264661604, 4.14991867284145974631861688589, 5.35684864602253773475513103494, 6.30298969327372768593273444330, 7.27315734999518814405234028609, 7.82838952228452500673863211994, 8.261635632931177593859400103507
