| L(s) = 1 | − 2.50·3-s + 5-s − 7-s + 3.29·9-s + 2.85·11-s + 13-s − 2.50·15-s + 4.55·17-s + 1.75·19-s + 2.50·21-s − 0.542·23-s + 25-s − 0.752·27-s + 0.0329·29-s + 0.890·31-s − 7.17·33-s − 35-s − 8.22·37-s − 2.50·39-s + 3.10·41-s + 0.800·43-s + 3.29·45-s − 12.1·47-s + 49-s − 11.4·51-s + 14.0·53-s + 2.85·55-s + ⋯ |
| L(s) = 1 | − 1.44·3-s + 0.447·5-s − 0.377·7-s + 1.09·9-s + 0.861·11-s + 0.277·13-s − 0.648·15-s + 1.10·17-s + 0.403·19-s + 0.547·21-s − 0.113·23-s + 0.200·25-s − 0.144·27-s + 0.00612·29-s + 0.159·31-s − 1.24·33-s − 0.169·35-s − 1.35·37-s − 0.401·39-s + 0.484·41-s + 0.122·43-s + 0.491·45-s − 1.77·47-s + 0.142·49-s − 1.60·51-s + 1.92·53-s + 0.385·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.216641053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216641053\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.50T + 3T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 17 | \( 1 - 4.55T + 17T^{2} \) |
| 19 | \( 1 - 1.75T + 19T^{2} \) |
| 23 | \( 1 + 0.542T + 23T^{2} \) |
| 29 | \( 1 - 0.0329T + 29T^{2} \) |
| 31 | \( 1 - 0.890T + 31T^{2} \) |
| 37 | \( 1 + 8.22T + 37T^{2} \) |
| 41 | \( 1 - 3.10T + 41T^{2} \) |
| 43 | \( 1 - 0.800T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 9.68T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 4.20T + 71T^{2} \) |
| 73 | \( 1 - 4.57T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 0.542T + 97T^{2} \) |
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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
−8.648067445954019999683990337381, −7.59660431817871974612541445126, −6.77939530116418026693441808517, −6.26516500331734798092205408540, −5.58378685365647000498197694528, −5.02795321845672948804606878939, −3.99770939223573634974440115926, −3.11933337009104152049129621195, −1.66383280932333377694101096228, −0.73192794264155785019801105878,
0.73192794264155785019801105878, 1.66383280932333377694101096228, 3.11933337009104152049129621195, 3.99770939223573634974440115926, 5.02795321845672948804606878939, 5.58378685365647000498197694528, 6.26516500331734798092205408540, 6.77939530116418026693441808517, 7.59660431817871974612541445126, 8.648067445954019999683990337381
