| L(s) = 1 | − 3·7-s − 5·11-s − 13-s − 5·17-s + 4·19-s + 2·23-s + 9·29-s + 3·31-s + 10·37-s + 12·41-s − 2·43-s + 9·47-s + 2·49-s + 9·53-s − 3·59-s − 7·61-s − 9·67-s + 10·73-s + 15·77-s − 10·79-s + 83-s + 4·89-s + 3·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 1.50·11-s − 0.277·13-s − 1.21·17-s + 0.917·19-s + 0.417·23-s + 1.67·29-s + 0.538·31-s + 1.64·37-s + 1.87·41-s − 0.304·43-s + 1.31·47-s + 2/7·49-s + 1.23·53-s − 0.390·59-s − 0.896·61-s − 1.09·67-s + 1.17·73-s + 1.70·77-s − 1.12·79-s + 0.109·83-s + 0.423·89-s + 0.314·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.543486378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.543486378\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−13.67515099118349, −13.37041647096323, −12.82151344425600, −12.52164000867877, −11.92838900583135, −11.31918379082559, −10.79814980491532, −10.31731727885154, −9.937645395836500, −9.332036216715277, −8.955089839982917, −8.297957296540214, −7.683629228972096, −7.327757051176766, −6.706609749415402, −6.097061081434653, −5.783150562912894, −4.935907522792931, −4.575640118265661, −3.926517937707714, −2.999459097193745, −2.716975243280124, −2.310830296367617, −1.039860023302283, −0.4519741229049433,
0.4519741229049433, 1.039860023302283, 2.310830296367617, 2.716975243280124, 2.999459097193745, 3.926517937707714, 4.575640118265661, 4.935907522792931, 5.783150562912894, 6.097061081434653, 6.706609749415402, 7.327757051176766, 7.683629228972096, 8.297957296540214, 8.955089839982917, 9.332036216715277, 9.937645395836500, 10.31731727885154, 10.79814980491532, 11.31918379082559, 11.92838900583135, 12.52164000867877, 12.82151344425600, 13.37041647096323, 13.67515099118349
