| L(s) = 1 | + 7-s + 6·11-s + 2·13-s − 4·19-s + 6·23-s − 5·25-s − 6·29-s + 8·31-s + 2·37-s − 12·41-s − 4·43-s − 12·47-s + 49-s + 6·53-s − 10·61-s + 8·67-s − 6·71-s − 10·73-s + 6·77-s − 4·79-s + 12·83-s − 12·89-s + 2·91-s − 10·97-s + 12·101-s + 8·103-s + 6·107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.80·11-s + 0.554·13-s − 0.917·19-s + 1.25·23-s − 25-s − 1.11·29-s + 1.43·31-s + 0.328·37-s − 1.87·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.824·53-s − 1.28·61-s + 0.977·67-s − 0.712·71-s − 1.17·73-s + 0.683·77-s − 0.450·79-s + 1.31·83-s − 1.27·89-s + 0.209·91-s − 1.01·97-s + 1.19·101-s + 0.788·103-s + 0.580·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.379796809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.379796809\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−11.78681575790007246498903350804, −11.34449628375814460891723631889, −10.10483686416197845980665091731, −9.072046417107109896038765861128, −8.300043091642042619777544931378, −6.93446262615281462668117301613, −6.11056271857714724112691707067, −4.63917458628061830491997962672, −3.53462971911322619523210724383, −1.58867103529813382887688476545,
1.58867103529813382887688476545, 3.53462971911322619523210724383, 4.63917458628061830491997962672, 6.11056271857714724112691707067, 6.93446262615281462668117301613, 8.300043091642042619777544931378, 9.072046417107109896038765861128, 10.10483686416197845980665091731, 11.34449628375814460891723631889, 11.78681575790007246498903350804
