| L(s) = 1 | − 11-s + 13-s + 6·17-s − 2·19-s + 6·23-s − 5·25-s + 9·29-s − 4·31-s − 2·37-s + 6·41-s + 4·43-s + 6·47-s − 3·59-s − 11·61-s − 11·67-s + 2·73-s + 5·79-s − 6·83-s + 18·89-s − 13·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s + 0.277·13-s + 1.45·17-s − 0.458·19-s + 1.25·23-s − 25-s + 1.67·29-s − 0.718·31-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 0.875·47-s − 0.390·59-s − 1.40·61-s − 1.34·67-s + 0.234·73-s + 0.562·79-s − 0.658·83-s + 1.90·89-s − 1.31·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.606058945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.606058945\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−12.45132148267132, −12.19454930492032, −12.01220600138651, −11.05683531988354, −10.88126265539370, −10.49294868873506, −9.813679313553658, −9.581108037868860, −8.838199254037718, −8.666575037870712, −7.866809527212245, −7.643797266842556, −7.182463840319141, −6.490636167560782, −6.077935458709234, −5.580338724492534, −5.129834419788081, −4.547860898963009, −4.023703902562940, −3.469608904940059, −2.858127031977022, −2.532385011492161, −1.606114188770207, −1.157635604276312, −0.4571056926853647,
0.4571056926853647, 1.157635604276312, 1.606114188770207, 2.532385011492161, 2.858127031977022, 3.469608904940059, 4.023703902562940, 4.547860898963009, 5.129834419788081, 5.580338724492534, 6.077935458709234, 6.490636167560782, 7.182463840319141, 7.643797266842556, 7.866809527212245, 8.666575037870712, 8.838199254037718, 9.581108037868860, 9.813679313553658, 10.49294868873506, 10.88126265539370, 11.05683531988354, 12.01220600138651, 12.19454930492032, 12.45132148267132
