| L(s) = 1 | − 2-s + 2.03·3-s + 4-s + 5-s − 2.03·6-s − 2.33·7-s − 8-s + 1.14·9-s − 10-s + 11-s + 2.03·12-s + 0.590·13-s + 2.33·14-s + 2.03·15-s + 16-s + 3.76·17-s − 1.14·18-s − 2.57·19-s + 20-s − 4.74·21-s − 22-s − 5.51·23-s − 2.03·24-s + 25-s − 0.590·26-s − 3.78·27-s − 2.33·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.17·3-s + 0.5·4-s + 0.447·5-s − 0.830·6-s − 0.880·7-s − 0.353·8-s + 0.380·9-s − 0.316·10-s + 0.301·11-s + 0.587·12-s + 0.163·13-s + 0.622·14-s + 0.525·15-s + 0.250·16-s + 0.914·17-s − 0.269·18-s − 0.591·19-s + 0.223·20-s − 1.03·21-s − 0.213·22-s − 1.14·23-s − 0.415·24-s + 0.200·25-s − 0.115·26-s − 0.727·27-s − 0.440·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 - 2.03T + 3T^{2} \) |
| 7 | \( 1 + 2.33T + 7T^{2} \) |
| 13 | \( 1 - 0.590T + 13T^{2} \) |
| 17 | \( 1 - 3.76T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 + 5.51T + 23T^{2} \) |
| 29 | \( 1 - 4.52T + 29T^{2} \) |
| 31 | \( 1 - 6.65T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 1.48T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 9.81T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 2.22T + 59T^{2} \) |
| 61 | \( 1 - 1.83T + 61T^{2} \) |
| 67 | \( 1 - 3.42T + 67T^{2} \) |
| 71 | \( 1 + 6.62T + 71T^{2} \) |
| 79 | \( 1 - 3.34T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 0.403T + 89T^{2} \) |
| 97 | \( 1 + 3.16T + 97T^{2} \) |
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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
−7.78911980617884780414489236918, −6.74490067073450633254918500481, −6.41719775583345625593498099094, −5.59504718117144104308973855353, −4.53693703665055971094169047359, −3.39328577955358738614162276880, −3.17716728309468293084910865716, −2.17694997159191777898703323553, −1.44901740124774815247512097148, 0,
1.44901740124774815247512097148, 2.17694997159191777898703323553, 3.17716728309468293084910865716, 3.39328577955358738614162276880, 4.53693703665055971094169047359, 5.59504718117144104308973855353, 6.41719775583345625593498099094, 6.74490067073450633254918500481, 7.78911980617884780414489236918
