| L(s) = 1 | + 2·3-s − 2·5-s + 7-s + 9-s − 2·11-s − 2·13-s − 4·15-s − 4·19-s + 2·21-s − 4·23-s − 25-s − 4·27-s − 2·29-s − 8·31-s − 4·33-s − 2·35-s + 2·37-s − 4·39-s − 4·43-s − 2·45-s + 12·47-s + 49-s − 14·53-s + 4·55-s − 8·57-s − 12·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 1.03·15-s − 0.917·19-s + 0.436·21-s − 0.834·23-s − 1/5·25-s − 0.769·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.338·35-s + 0.328·37-s − 0.640·39-s − 0.609·43-s − 0.298·45-s + 1.75·47-s + 1/7·49-s − 1.92·53-s + 0.539·55-s − 1.05·57-s − 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.200800035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.200800035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−15.02123693296786, −14.44907830473147, −14.24471242844370, −13.46548439813770, −13.02960563004189, −12.42565494775902, −11.93626879518887, −11.29223304010648, −10.81364072917071, −10.21993841805036, −9.520518339694775, −9.008377126875359, −8.489160750297061, −7.881998018660280, −7.645313697720401, −7.142635020808768, −6.152463052868837, −5.603929069100757, −4.725267974285974, −4.233201187751698, −3.573477965951280, −3.058857926708224, −2.173187230716611, −1.836015796499907, −0.3584623519317678,
0.3584623519317678, 1.836015796499907, 2.173187230716611, 3.058857926708224, 3.573477965951280, 4.233201187751698, 4.725267974285974, 5.603929069100757, 6.152463052868837, 7.142635020808768, 7.645313697720401, 7.881998018660280, 8.489160750297061, 9.008377126875359, 9.520518339694775, 10.21993841805036, 10.81364072917071, 11.29223304010648, 11.93626879518887, 12.42565494775902, 13.02960563004189, 13.46548439813770, 14.24471242844370, 14.44907830473147, 15.02123693296786
