| L(s) = 1 | + 2-s − 3.38·3-s + 4-s + 5-s − 3.38·6-s + 1.27·7-s + 8-s + 8.49·9-s + 10-s + 11-s − 3.38·12-s − 6.37·13-s + 1.27·14-s − 3.38·15-s + 16-s − 4.31·17-s + 8.49·18-s + 2.03·19-s + 20-s − 4.32·21-s + 22-s + 5.87·23-s − 3.38·24-s + 25-s − 6.37·26-s − 18.6·27-s + 1.27·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.95·3-s + 0.5·4-s + 0.447·5-s − 1.38·6-s + 0.482·7-s + 0.353·8-s + 2.83·9-s + 0.316·10-s + 0.301·11-s − 0.978·12-s − 1.76·13-s + 0.341·14-s − 0.875·15-s + 0.250·16-s − 1.04·17-s + 2.00·18-s + 0.467·19-s + 0.223·20-s − 0.943·21-s + 0.213·22-s + 1.22·23-s − 0.691·24-s + 0.200·25-s − 1.24·26-s − 3.58·27-s + 0.241·28-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 | \( 1 + 3.38T + 3T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 13 | \( 1 + 6.37T + 13T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 19 | \( 1 - 2.03T + 19T^{2} \) |
| 23 | \( 1 - 5.87T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 - 8.83T + 31T^{2} \) |
| 37 | \( 1 + 1.21T + 37T^{2} \) |
| 41 | \( 1 - 3.71T + 41T^{2} \) |
| 43 | \( 1 + 6.71T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 8.27T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 0.979T + 61T^{2} \) |
| 67 | \( 1 - 0.367T + 67T^{2} \) |
| 71 | \( 1 + 4.19T + 71T^{2} \) |
| 79 | \( 1 + 7.97T + 79T^{2} \) |
| 83 | \( 1 + 9.52T + 83T^{2} \) |
| 89 | \( 1 - 18.4T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−7.22698924985293035783224802745, −6.44608969972375804424223704341, −6.20460651451581670890250914475, −5.14261691012585176670103438360, −4.77736178699760093240364678701, −4.56882341771359168587961538673, −3.18418886463687008291857568767, −2.06397469842159815832364407414, −1.23659710041538473479848814616, 0,
1.23659710041538473479848814616, 2.06397469842159815832364407414, 3.18418886463687008291857568767, 4.56882341771359168587961538673, 4.77736178699760093240364678701, 5.14261691012585176670103438360, 6.20460651451581670890250914475, 6.44608969972375804424223704341, 7.22698924985293035783224802745
