| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s + 6·11-s + 13-s − 15-s − 17-s + 6·19-s + 4·21-s + 25-s + 27-s − 6·29-s + 4·31-s + 6·33-s − 4·35-s + 4·37-s + 39-s + 2·41-s − 4·43-s − 45-s + 9·49-s − 51-s − 2·53-s − 6·55-s + 6·57-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.80·11-s + 0.277·13-s − 0.258·15-s − 0.242·17-s + 1.37·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s + 1.04·33-s − 0.676·35-s + 0.657·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s − 0.149·45-s + 9/7·49-s − 0.140·51-s − 0.274·53-s − 0.809·55-s + 0.794·57-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.214915024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.214915024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.10036997990143, −12.36490251873824, −11.99154631541181, −11.59590098461310, −11.18453867686767, −10.94877956432684, −10.09746228597477, −9.564312049390812, −9.235847953258166, −8.682157272470207, −8.324681047947170, −7.778582562462341, −7.403307595036672, −6.934231190572539, −6.297665158925677, −5.774829160890831, −5.082011842946677, −4.567471425153525, −4.221738874049231, −3.514264847232919, −3.289721027616534, −2.277063494952477, −1.789469725558905, −1.194257968470780, −0.7541569843247099,
0.7541569843247099, 1.194257968470780, 1.789469725558905, 2.277063494952477, 3.289721027616534, 3.514264847232919, 4.221738874049231, 4.567471425153525, 5.082011842946677, 5.774829160890831, 6.297665158925677, 6.934231190572539, 7.403307595036672, 7.778582562462341, 8.324681047947170, 8.682157272470207, 9.235847953258166, 9.564312049390812, 10.09746228597477, 10.94877956432684, 11.18453867686767, 11.59590098461310, 11.99154631541181, 12.36490251873824, 13.10036997990143
