| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 15-s + 3·17-s + 6·19-s − 21-s + 4·23-s − 4·25-s + 27-s − 2·29-s + 10·31-s + 35-s − 2·37-s + 10·41-s − 3·43-s − 45-s + 9·47-s + 49-s + 3·51-s + 6·57-s − 11·59-s − 63-s − 5·67-s + 4·69-s + 16·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.258·15-s + 0.727·17-s + 1.37·19-s − 0.218·21-s + 0.834·23-s − 4/5·25-s + 0.192·27-s − 0.371·29-s + 1.79·31-s + 0.169·35-s − 0.328·37-s + 1.56·41-s − 0.457·43-s − 0.149·45-s + 1.31·47-s + 1/7·49-s + 0.420·51-s + 0.794·57-s − 1.43·59-s − 0.125·63-s − 0.610·67-s + 0.481·69-s + 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.710090227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.710090227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.73801070705691, −15.20582256637137, −14.47392976182737, −14.02007232879234, −13.53234279063765, −13.01723632200710, −12.11306292165820, −12.09014668205684, −11.24560670341323, −10.64886175580804, −9.918339471881707, −9.539020159110727, −8.976655728149734, −8.263241852113372, −7.634087151812083, −7.372336583578896, −6.547254137131626, −5.841664872843142, −5.203214521457458, −4.402470107522565, −3.783081726396214, −3.062409800851454, −2.634215499492529, −1.463861289054988, −0.6979839094145961,
0.6979839094145961, 1.463861289054988, 2.634215499492529, 3.062409800851454, 3.783081726396214, 4.402470107522565, 5.203214521457458, 5.841664872843142, 6.547254137131626, 7.372336583578896, 7.634087151812083, 8.263241852113372, 8.976655728149734, 9.539020159110727, 9.918339471881707, 10.64886175580804, 11.24560670341323, 12.09014668205684, 12.11306292165820, 13.01723632200710, 13.53234279063765, 14.02007232879234, 14.47392976182737, 15.20582256637137, 15.73801070705691
