| L(s) = 1 | − 2.41·2-s + 2.99·3-s + 3.80·4-s − 2.37·5-s − 7.21·6-s + 3.58·7-s − 4.36·8-s + 5.95·9-s + 5.72·10-s − 5.47·11-s + 11.3·12-s + 2.68·13-s − 8.63·14-s − 7.10·15-s + 2.89·16-s + 4.30·17-s − 14.3·18-s + 3.21·19-s − 9.04·20-s + 10.7·21-s + 13.1·22-s − 2.38·23-s − 13.0·24-s + 0.638·25-s − 6.47·26-s + 8.83·27-s + 13.6·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.72·3-s + 1.90·4-s − 1.06·5-s − 2.94·6-s + 1.35·7-s − 1.54·8-s + 1.98·9-s + 1.80·10-s − 1.65·11-s + 3.29·12-s + 0.744·13-s − 2.30·14-s − 1.83·15-s + 0.723·16-s + 1.04·17-s − 3.38·18-s + 0.737·19-s − 2.02·20-s + 2.34·21-s + 2.81·22-s − 0.497·23-s − 2.66·24-s + 0.127·25-s − 1.26·26-s + 1.70·27-s + 2.58·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3467 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3467 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.540117060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.540117060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3467 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 - 2.99T + 3T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 - 3.58T + 7T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 - 2.68T + 13T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 19 | \( 1 - 3.21T + 19T^{2} \) |
| 23 | \( 1 + 2.38T + 23T^{2} \) |
| 29 | \( 1 - 1.63T + 29T^{2} \) |
| 31 | \( 1 - 0.794T + 31T^{2} \) |
| 37 | \( 1 + 5.49T + 37T^{2} \) |
| 41 | \( 1 - 2.65T + 41T^{2} \) |
| 43 | \( 1 - 7.32T + 43T^{2} \) |
| 47 | \( 1 - 6.93T + 47T^{2} \) |
| 53 | \( 1 - 0.702T + 53T^{2} \) |
| 59 | \( 1 - 9.07T + 59T^{2} \) |
| 61 | \( 1 - 6.59T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 - 6.51T + 73T^{2} \) |
| 79 | \( 1 - 1.24T + 79T^{2} \) |
| 83 | \( 1 - 0.923T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 8.34T + 97T^{2} \) |
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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
−8.398167331652553475568747961471, −8.079447840911563276957978819778, −7.46766486379510519925144056044, −7.36619086173507560920295641632, −5.65812253744150070332697038649, −4.53065132790897126483994595434, −3.56759936060765293866436990341, −2.70647365363733414443867209359, −1.90684870836039067759847736430, −0.903011764759659985844171194719,
0.903011764759659985844171194719, 1.90684870836039067759847736430, 2.70647365363733414443867209359, 3.56759936060765293866436990341, 4.53065132790897126483994595434, 5.65812253744150070332697038649, 7.36619086173507560920295641632, 7.46766486379510519925144056044, 8.079447840911563276957978819778, 8.398167331652553475568747961471
