| L(s) = 1 | − 3-s − 3·5-s − 4·7-s − 2·9-s − 3·11-s − 5·13-s + 3·15-s − 6·17-s + 4·19-s + 4·21-s − 6·23-s + 4·25-s + 5·27-s + 29-s + 5·31-s + 3·33-s + 12·35-s − 8·37-s + 5·39-s + 43-s + 6·45-s − 3·47-s + 9·49-s + 6·51-s − 3·53-s + 9·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 1.51·7-s − 2/3·9-s − 0.904·11-s − 1.38·13-s + 0.774·15-s − 1.45·17-s + 0.917·19-s + 0.872·21-s − 1.25·23-s + 4/5·25-s + 0.962·27-s + 0.185·29-s + 0.898·31-s + 0.522·33-s + 2.02·35-s − 1.31·37-s + 0.800·39-s + 0.152·43-s + 0.894·45-s − 0.437·47-s + 9/7·49-s + 0.840·51-s − 0.412·53-s + 1.21·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.365849460230947911080889695766, −7.59809898201067299620356606196, −6.88153422745591666330204489645, −6.12873837822252423292472846453, −5.14212253637157501031773520025, −4.34009024089584088961912752686, −3.25826208536459815397477851091, −2.57274953389751390703429915549, 0, 0,
2.57274953389751390703429915549, 3.25826208536459815397477851091, 4.34009024089584088961912752686, 5.14212253637157501031773520025, 6.12873837822252423292472846453, 6.88153422745591666330204489645, 7.59809898201067299620356606196, 8.365849460230947911080889695766
