| L(s) = 1 | − 0.589·3-s + 1.60·5-s − 7-s − 2.65·9-s − 5.45·11-s − 6.59·13-s − 0.943·15-s + 5.33·17-s − 3.61·19-s + 0.589·21-s + 2.60·23-s − 2.43·25-s + 3.33·27-s − 1.72·29-s + 0.833·31-s + 3.21·33-s − 1.60·35-s + 6.26·37-s + 3.88·39-s − 0.263·41-s − 1.77·43-s − 4.24·45-s − 10.7·47-s + 49-s − 3.14·51-s + 0.0673·53-s − 8.73·55-s + ⋯ |
| L(s) = 1 | − 0.340·3-s + 0.715·5-s − 0.377·7-s − 0.884·9-s − 1.64·11-s − 1.82·13-s − 0.243·15-s + 1.29·17-s − 0.830·19-s + 0.128·21-s + 0.543·23-s − 0.487·25-s + 0.641·27-s − 0.321·29-s + 0.149·31-s + 0.559·33-s − 0.270·35-s + 1.02·37-s + 0.622·39-s − 0.0411·41-s − 0.270·43-s − 0.632·45-s − 1.56·47-s + 0.142·49-s − 0.440·51-s + 0.00925·53-s − 1.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7874281803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7874281803\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 0.589T + 3T^{2} \) |
| 5 | \( 1 - 1.60T + 5T^{2} \) |
| 11 | \( 1 + 5.45T + 11T^{2} \) |
| 13 | \( 1 + 6.59T + 13T^{2} \) |
| 17 | \( 1 - 5.33T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 - 2.60T + 23T^{2} \) |
| 29 | \( 1 + 1.72T + 29T^{2} \) |
| 31 | \( 1 - 0.833T + 31T^{2} \) |
| 37 | \( 1 - 6.26T + 37T^{2} \) |
| 41 | \( 1 + 0.263T + 41T^{2} \) |
| 43 | \( 1 + 1.77T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 0.0673T + 53T^{2} \) |
| 59 | \( 1 + 5.10T + 59T^{2} \) |
| 61 | \( 1 - 6.31T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 2.05T + 71T^{2} \) |
| 73 | \( 1 + 5.48T + 73T^{2} \) |
| 79 | \( 1 - 5.21T + 79T^{2} \) |
| 83 | \( 1 - 8.25T + 83T^{2} \) |
| 89 | \( 1 - 6.32T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79739521759947425751963663045, −7.37712319836262580633589258541, −6.33484817707066659517891522130, −5.79131026022293022662165584489, −5.13966192228376111153672973175, −4.71536622222765937176802457845, −3.27753484427190609588046819367, −2.68591408161055597650498424708, −2.01824029978116048712880775911, −0.42348852738371342122919420473,
0.42348852738371342122919420473, 2.01824029978116048712880775911, 2.68591408161055597650498424708, 3.27753484427190609588046819367, 4.71536622222765937176802457845, 5.13966192228376111153672973175, 5.79131026022293022662165584489, 6.33484817707066659517891522130, 7.37712319836262580633589258541, 7.79739521759947425751963663045
