| L(s) = 1 | + 5-s + 2·7-s + 2·11-s + 6·13-s − 4·17-s − 4·19-s + 25-s + 2·29-s − 4·31-s + 2·35-s − 8·37-s + 8·41-s − 43-s + 8·47-s − 3·49-s − 2·53-s + 2·55-s − 6·59-s + 4·61-s + 6·65-s + 12·67-s + 10·73-s + 4·77-s + 8·79-s + 14·83-s − 4·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.603·11-s + 1.66·13-s − 0.970·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s − 0.718·31-s + 0.338·35-s − 1.31·37-s + 1.24·41-s − 0.152·43-s + 1.16·47-s − 3/7·49-s − 0.274·53-s + 0.269·55-s − 0.781·59-s + 0.512·61-s + 0.744·65-s + 1.46·67-s + 1.17·73-s + 0.455·77-s + 0.900·79-s + 1.53·83-s − 0.433·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.713678885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.713678885\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.53919052183250, −13.18315961228930, −12.50194678716046, −12.22656532489319, −11.37267517720767, −11.11012566260431, −10.73648298710910, −10.34139542894099, −9.408708275180701, −9.164057341860126, −8.587270593074921, −8.308665205518251, −7.676989759350247, −6.955798825391146, −6.434562413474815, −6.185198391623549, −5.479877405325959, −4.939953751399407, −4.316887740070880, −3.818185898038450, −3.345425199351206, −2.312839508498387, −1.961926914648670, −1.295625947466621, −0.6083067469353147,
0.6083067469353147, 1.295625947466621, 1.961926914648670, 2.312839508498387, 3.345425199351206, 3.818185898038450, 4.316887740070880, 4.939953751399407, 5.479877405325959, 6.185198391623549, 6.434562413474815, 6.955798825391146, 7.676989759350247, 8.308665205518251, 8.587270593074921, 9.164057341860126, 9.408708275180701, 10.34139542894099, 10.73648298710910, 11.11012566260431, 11.37267517720767, 12.22656532489319, 12.50194678716046, 13.18315961228930, 13.53919052183250
