| L(s) = 1 | + 21.8·7-s − 54.6·11-s + 82.7·13-s + 100.·17-s − 84.1·19-s + 0.880·23-s + 99.1·29-s − 78.9·31-s + 390.·37-s − 104.·41-s + 241.·43-s − 512.·47-s + 135.·49-s − 284.·53-s + 709.·59-s + 470.·61-s − 667.·67-s + 51.5·71-s + 371.·73-s − 1.19e3·77-s − 79.3·79-s + 682.·83-s − 628.·89-s + 1.81e3·91-s + 1.51e3·97-s − 977.·101-s + 759.·103-s + ⋯ |
| L(s) = 1 | + 1.18·7-s − 1.49·11-s + 1.76·13-s + 1.43·17-s − 1.01·19-s + 0.00798·23-s + 0.634·29-s − 0.457·31-s + 1.73·37-s − 0.398·41-s + 0.856·43-s − 1.59·47-s + 0.395·49-s − 0.737·53-s + 1.56·59-s + 0.987·61-s − 1.21·67-s + 0.0861·71-s + 0.595·73-s − 1.76·77-s − 0.112·79-s + 0.902·83-s − 0.748·89-s + 2.08·91-s + 1.59·97-s − 0.963·101-s + 0.726·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.689987278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.689987278\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 21.8T + 343T^{2} \) |
| 11 | \( 1 + 54.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 82.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 100.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 84.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 0.880T + 1.21e4T^{2} \) |
| 29 | \( 1 - 99.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 78.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 390.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 104.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 241.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 512.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 284.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 709.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 470.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 667.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 51.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 371.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 79.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 682.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 628.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.51e3T + 9.12e5T^{2} \) |
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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
−8.612473651858141756793260078338, −8.127055496151175834084902408213, −7.65477337863727619094425209220, −6.39770078455711249732877458970, −5.62882237395372288262894014377, −4.90817660325009872445275564734, −3.95293909880340057119330333349, −2.91640690038644858667662102210, −1.79554977295481443290932965257, −0.805384742048128437012888282507,
0.805384742048128437012888282507, 1.79554977295481443290932965257, 2.91640690038644858667662102210, 3.95293909880340057119330333349, 4.90817660325009872445275564734, 5.62882237395372288262894014377, 6.39770078455711249732877458970, 7.65477337863727619094425209220, 8.127055496151175834084902408213, 8.612473651858141756793260078338
