| L(s) = 1 | − 3-s + 4·5-s + 9-s − 11-s − 13-s − 4·15-s + 17-s − 2·19-s + 4·23-s + 11·25-s − 27-s + 2·29-s + 33-s − 6·37-s + 39-s + 10·41-s + 6·43-s + 4·45-s + 10·47-s − 7·49-s − 51-s − 6·53-s − 4·55-s + 2·57-s − 14·59-s + 8·61-s − 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 1.03·15-s + 0.242·17-s − 0.458·19-s + 0.834·23-s + 11/5·25-s − 0.192·27-s + 0.371·29-s + 0.174·33-s − 0.986·37-s + 0.160·39-s + 1.56·41-s + 0.914·43-s + 0.596·45-s + 1.45·47-s − 49-s − 0.140·51-s − 0.824·53-s − 0.539·55-s + 0.264·57-s − 1.82·59-s + 1.02·61-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.95573139927338, −13.23222257013944, −12.86445457079989, −12.55808674071220, −12.01653141408360, −11.25372866317888, −10.82164274613882, −10.40353175810711, −10.05750945388709, −9.433372373180698, −9.062106818664957, −8.658224543380063, −7.745545068177488, −7.318402118816629, −6.717008598999092, −6.132134252554602, −5.881520520058750, −5.329822354108760, −4.789544026951673, −4.364305803888655, −3.391460265760784, −2.685347253476584, −2.313070298671550, −1.488785705406133, −1.061456839045156, 0,
1.061456839045156, 1.488785705406133, 2.313070298671550, 2.685347253476584, 3.391460265760784, 4.364305803888655, 4.789544026951673, 5.329822354108760, 5.881520520058750, 6.132134252554602, 6.717008598999092, 7.318402118816629, 7.745545068177488, 8.658224543380063, 9.062106818664957, 9.433372373180698, 10.05750945388709, 10.40353175810711, 10.82164274613882, 11.25372866317888, 12.01653141408360, 12.55808674071220, 12.86445457079989, 13.23222257013944, 13.95573139927338
