L(s) = 1 | + 5-s + 4·7-s − 3·9-s − 11-s + 6·13-s − 6·17-s + 4·19-s + 4·23-s + 25-s − 2·29-s + 8·31-s + 4·35-s − 10·37-s + 10·41-s − 3·45-s + 4·47-s + 9·49-s − 10·53-s − 55-s − 4·59-s − 2·61-s − 12·63-s + 6·65-s − 8·67-s − 14·73-s − 4·77-s − 16·79-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.51·7-s − 9-s − 0.301·11-s + 1.66·13-s − 1.45·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.676·35-s − 1.64·37-s + 1.56·41-s − 0.447·45-s + 0.583·47-s + 9/7·49-s − 1.37·53-s − 0.134·55-s − 0.520·59-s − 0.256·61-s − 1.51·63-s + 0.744·65-s − 0.977·67-s − 1.63·73-s − 0.455·77-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.676754790\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.676754790\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−11.19535799739602312775645550347, −10.51644634403824787312406166098, −8.951396551109647722305086825963, −8.613808585244655340820053101754, −7.58115128738550044456594424531, −6.28118370578645402196158692333, −5.40941065075534095194244033749, −4.42524465686514871694159803405, −2.88180193076351113983639438164, −1.46303061533755639665183373410,
1.46303061533755639665183373410, 2.88180193076351113983639438164, 4.42524465686514871694159803405, 5.40941065075534095194244033749, 6.28118370578645402196158692333, 7.58115128738550044456594424531, 8.613808585244655340820053101754, 8.951396551109647722305086825963, 10.51644634403824787312406166098, 11.19535799739602312775645550347