| L(s) = 1 | + 6.32·3-s + 18.9·7-s + 13.0·9-s + 12.6·11-s − 38·13-s − 34·17-s + 101.·19-s + 120.·21-s + 82.2·23-s − 88.5·27-s + 270·29-s + 341.·31-s + 80.0·33-s − 206·37-s − 240.·39-s − 270·41-s + 537.·43-s − 132.·47-s + 17·49-s − 215.·51-s + 258·53-s + 640.·57-s + 75.8·59-s − 250·61-s + 246.·63-s − 815.·67-s + 520·69-s + ⋯ |
| L(s) = 1 | + 1.21·3-s + 1.02·7-s + 0.481·9-s + 0.346·11-s − 0.810·13-s − 0.485·17-s + 1.22·19-s + 1.24·21-s + 0.745·23-s − 0.631·27-s + 1.72·29-s + 1.97·31-s + 0.422·33-s − 0.915·37-s − 0.986·39-s − 1.02·41-s + 1.90·43-s − 0.412·47-s + 0.0495·49-s − 0.590·51-s + 0.668·53-s + 1.48·57-s + 0.167·59-s − 0.524·61-s + 0.493·63-s − 1.48·67-s + 0.907·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.740002685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.740002685\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 6.32T + 27T^{2} \) |
| 7 | \( 1 - 18.9T + 343T^{2} \) |
| 11 | \( 1 - 12.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38T + 2.19e3T^{2} \) |
| 17 | \( 1 + 34T + 4.91e3T^{2} \) |
| 19 | \( 1 - 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 82.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 270T + 2.43e4T^{2} \) |
| 31 | \( 1 - 341.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 206T + 5.06e4T^{2} \) |
| 41 | \( 1 + 270T + 6.89e4T^{2} \) |
| 43 | \( 1 - 537.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 132.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 258T + 1.48e5T^{2} \) |
| 59 | \( 1 - 75.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 250T + 2.26e5T^{2} \) |
| 67 | \( 1 + 815.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 645.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 278.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 890T + 7.04e5T^{2} \) |
| 97 | \( 1 - 254T + 9.12e5T^{2} \) |
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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
−9.701997660618840758960782394623, −8.900962433995662839984440889367, −8.231376333426647150968870971040, −7.55056901872421358901105903769, −6.60619216345694006662681261185, −5.17645615559185625847641613917, −4.43395999108753006251037357154, −3.15688626071505511407028331992, −2.35348824279759641073464597163, −1.09796248706382642042406981247,
1.09796248706382642042406981247, 2.35348824279759641073464597163, 3.15688626071505511407028331992, 4.43395999108753006251037357154, 5.17645615559185625847641613917, 6.60619216345694006662681261185, 7.55056901872421358901105903769, 8.231376333426647150968870971040, 8.900962433995662839984440889367, 9.701997660618840758960782394623
