| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 2·11-s − 13-s − 14-s + 16-s + 3·17-s + 6·19-s + 20-s − 2·22-s + 4·23-s − 4·25-s + 26-s + 28-s − 2·29-s + 4·31-s − 32-s − 3·34-s + 35-s + 3·37-s − 6·38-s − 40-s − 5·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.603·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 1.37·19-s + 0.223·20-s − 0.426·22-s + 0.834·23-s − 4/5·25-s + 0.196·26-s + 0.188·28-s − 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s + 0.169·35-s + 0.493·37-s − 0.973·38-s − 0.158·40-s − 0.762·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.041572906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.041572906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−11.86689967980337109441141028229, −11.27108277800122947493852923485, −9.945515510046352754787747239151, −9.474403868218177338841530557687, −8.251376244266523722285240612658, −7.34105417162701727305699109547, −6.18070949711225184510895261552, −4.98797766425562785919630567490, −3.20294866149791344283024452913, −1.48472360333277511919704091526,
1.48472360333277511919704091526, 3.20294866149791344283024452913, 4.98797766425562785919630567490, 6.18070949711225184510895261552, 7.34105417162701727305699109547, 8.251376244266523722285240612658, 9.474403868218177338841530557687, 9.945515510046352754787747239151, 11.27108277800122947493852923485, 11.86689967980337109441141028229
