| L(s) = 1 | − 3-s − 2.48·5-s − 7-s + 9-s − 5.58·11-s − 5.15·13-s + 2.48·15-s + 3.97·17-s − 5.58·19-s + 21-s − 23-s + 1.18·25-s − 27-s − 1.84·29-s − 8.60·31-s + 5.58·33-s + 2.48·35-s + 2.80·37-s + 5.15·39-s − 4.16·41-s + 4.11·43-s − 2.48·45-s − 4.18·47-s + 49-s − 3.97·51-s − 8.87·53-s + 13.8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.11·5-s − 0.377·7-s + 0.333·9-s − 1.68·11-s − 1.42·13-s + 0.641·15-s + 0.964·17-s − 1.28·19-s + 0.218·21-s − 0.208·23-s + 0.236·25-s − 0.192·27-s − 0.342·29-s − 1.54·31-s + 0.972·33-s + 0.420·35-s + 0.460·37-s + 0.825·39-s − 0.650·41-s + 0.627·43-s − 0.370·45-s − 0.611·47-s + 0.142·49-s − 0.556·51-s − 1.21·53-s + 1.87·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1840097383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1840097383\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2.48T + 5T^{2} \) |
| 11 | \( 1 + 5.58T + 11T^{2} \) |
| 13 | \( 1 + 5.15T + 13T^{2} \) |
| 17 | \( 1 - 3.97T + 17T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 29 | \( 1 + 1.84T + 29T^{2} \) |
| 31 | \( 1 + 8.60T + 31T^{2} \) |
| 37 | \( 1 - 2.80T + 37T^{2} \) |
| 41 | \( 1 + 4.16T + 41T^{2} \) |
| 43 | \( 1 - 4.11T + 43T^{2} \) |
| 47 | \( 1 + 4.18T + 47T^{2} \) |
| 53 | \( 1 + 8.87T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 + 6.89T + 61T^{2} \) |
| 67 | \( 1 + 7.59T + 67T^{2} \) |
| 71 | \( 1 - 5.88T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 5.08T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
| 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
−8.132086530712010346528122703341, −7.74486625433473356120211702301, −7.21271768873949686445117848645, −6.25084312540621514062346881072, −5.34852785872648859773137874678, −4.82323579554260020173765768774, −3.91838058503412446583292277458, −3.03033486058376197145601765755, −2.05874050617516653034354634267, −0.23686471058782269066763689600,
0.23686471058782269066763689600, 2.05874050617516653034354634267, 3.03033486058376197145601765755, 3.91838058503412446583292277458, 4.82323579554260020173765768774, 5.34852785872648859773137874678, 6.25084312540621514062346881072, 7.21271768873949686445117848645, 7.74486625433473356120211702301, 8.132086530712010346528122703341
