| L(s) = 1 | + 6·7-s + 19·11-s − 12·13-s − 75·17-s − 91·19-s + 174·23-s + 272·29-s − 230·31-s + 182·37-s − 117·41-s − 372·43-s − 52·47-s − 307·49-s − 402·53-s − 312·59-s + 170·61-s − 763·67-s + 52·71-s + 981·73-s + 114·77-s + 1.05e3·79-s + 351·83-s − 799·89-s − 72·91-s − 962·97-s − 486·101-s + 1.18e3·103-s + ⋯ |
| L(s) = 1 | + 0.323·7-s + 0.520·11-s − 0.256·13-s − 1.07·17-s − 1.09·19-s + 1.57·23-s + 1.74·29-s − 1.33·31-s + 0.808·37-s − 0.445·41-s − 1.31·43-s − 0.161·47-s − 0.895·49-s − 1.04·53-s − 0.688·59-s + 0.356·61-s − 1.39·67-s + 0.0869·71-s + 1.57·73-s + 0.168·77-s + 1.50·79-s + 0.464·83-s − 0.951·89-s − 0.0829·91-s − 1.00·97-s − 0.478·101-s + 1.13·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 19 T + p^{3} T^{2} \) |
| 13 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 + 75 T + p^{3} T^{2} \) |
| 19 | \( 1 + 91 T + p^{3} T^{2} \) |
| 23 | \( 1 - 174 T + p^{3} T^{2} \) |
| 29 | \( 1 - 272 T + p^{3} T^{2} \) |
| 31 | \( 1 + 230 T + p^{3} T^{2} \) |
| 37 | \( 1 - 182 T + p^{3} T^{2} \) |
| 41 | \( 1 + 117 T + p^{3} T^{2} \) |
| 43 | \( 1 + 372 T + p^{3} T^{2} \) |
| 47 | \( 1 + 52 T + p^{3} T^{2} \) |
| 53 | \( 1 + 402 T + p^{3} T^{2} \) |
| 59 | \( 1 + 312 T + p^{3} T^{2} \) |
| 61 | \( 1 - 170 T + p^{3} T^{2} \) |
| 67 | \( 1 + 763 T + p^{3} T^{2} \) |
| 71 | \( 1 - 52 T + p^{3} T^{2} \) |
| 73 | \( 1 - 981 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1054 T + p^{3} T^{2} \) |
| 83 | \( 1 - 351 T + p^{3} T^{2} \) |
| 89 | \( 1 + 799 T + p^{3} T^{2} \) |
| 97 | \( 1 + 962 T + p^{3} T^{2} \) |
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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
−8.634443284556281649619902805354, −7.78989065459425555050579537931, −6.73657000159277879210805803923, −6.38626351053244052214073955279, −5.03322033052604353820124937080, −4.54325313026006940833695880846, −3.42885600185247222911868561211, −2.37418120302380576142940113384, −1.33328183603611817257975340357, 0,
1.33328183603611817257975340357, 2.37418120302380576142940113384, 3.42885600185247222911868561211, 4.54325313026006940833695880846, 5.03322033052604353820124937080, 6.38626351053244052214073955279, 6.73657000159277879210805803923, 7.78989065459425555050579537931, 8.634443284556281649619902805354
