| L(s) = 1 | − 2·4-s − 5·7-s − 5·13-s + 4·16-s − 19-s + 10·28-s − 7·31-s + 10·37-s − 5·43-s + 18·49-s + 10·52-s − 13·61-s − 8·64-s − 5·67-s + 10·73-s + 2·76-s − 4·79-s + 25·91-s − 5·97-s − 20·103-s − 19·109-s − 20·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.88·7-s − 1.38·13-s + 16-s − 0.229·19-s + 1.88·28-s − 1.25·31-s + 1.64·37-s − 0.762·43-s + 18/7·49-s + 1.38·52-s − 1.66·61-s − 64-s − 0.610·67-s + 1.17·73-s + 0.229·76-s − 0.450·79-s + 2.62·91-s − 0.507·97-s − 1.97·103-s − 1.81·109-s − 1.88·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−12.11951399488920175322797663822, −10.49393585601286522315094653400, −9.599576350131101649309623215930, −9.206590206514279856778701903578, −7.75954149302713039814888795826, −6.62145632893118011347307968766, −5.46981299804941090347456591942, −4.14585724745469274814887485348, −2.93248946372060750659719660133, 0,
2.93248946372060750659719660133, 4.14585724745469274814887485348, 5.46981299804941090347456591942, 6.62145632893118011347307968766, 7.75954149302713039814888795826, 9.206590206514279856778701903578, 9.599576350131101649309623215930, 10.49393585601286522315094653400, 12.11951399488920175322797663822
