| L(s) = 1 | + 3-s + 7-s + 9-s + 2·11-s + 6·13-s − 8·17-s − 19-s + 21-s + 2·23-s − 5·25-s + 27-s + 6·29-s + 2·33-s + 10·37-s + 6·39-s − 8·41-s + 12·43-s − 4·47-s + 49-s − 8·51-s + 2·53-s − 57-s + 8·59-s − 10·61-s + 63-s + 12·67-s + 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.66·13-s − 1.94·17-s − 0.229·19-s + 0.218·21-s + 0.417·23-s − 25-s + 0.192·27-s + 1.11·29-s + 0.348·33-s + 1.64·37-s + 0.960·39-s − 1.24·41-s + 1.82·43-s − 0.583·47-s + 1/7·49-s − 1.12·51-s + 0.274·53-s − 0.132·57-s + 1.04·59-s − 1.28·61-s + 0.125·63-s + 1.46·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.957495443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.957495443\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−8.246755489210352119551965106673, −7.37964553234722343293322585702, −6.41789992631738856012763131277, −6.25830190106659993628925336325, −5.03817203960855039258796011079, −4.18277348693860540612893112565, −3.80510044018255037298785242872, −2.67985419265400239936455569293, −1.89040327267883459303388156147, −0.905132273914046520704532408679,
0.905132273914046520704532408679, 1.89040327267883459303388156147, 2.67985419265400239936455569293, 3.80510044018255037298785242872, 4.18277348693860540612893112565, 5.03817203960855039258796011079, 6.25830190106659993628925336325, 6.41789992631738856012763131277, 7.37964553234722343293322585702, 8.246755489210352119551965106673
