| L(s) = 1 | + 11·7-s + 23·13-s − 37·19-s + 25·25-s − 46·31-s − 73·37-s − 22·43-s + 72·49-s + 47·61-s − 13·67-s + 143·73-s + 11·79-s + 253·91-s − 169·97-s − 157·103-s − 214·109-s + ⋯ |
| L(s) = 1 | + 11/7·7-s + 1.76·13-s − 1.94·19-s + 25-s − 1.48·31-s − 1.97·37-s − 0.511·43-s + 1.46·49-s + 0.770·61-s − 0.194·67-s + 1.95·73-s + 0.139·79-s + 2.78·91-s − 1.74·97-s − 1.52·103-s − 1.96·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.547173766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.547173766\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 - 11 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 23 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 37 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 46 T + p^{2} T^{2} \) |
| 37 | \( 1 + 73 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 22 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 47 T + p^{2} T^{2} \) |
| 67 | \( 1 + 13 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 143 T + p^{2} T^{2} \) |
| 79 | \( 1 - 11 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 169 T + p^{2} T^{2} \) |
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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
−13.53445154385831040337638587948, −12.42697619570053259749815883598, −11.06117180223364224220707158157, −10.74228228770574286361146919753, −8.788020066621735944813417079021, −8.247366710731016436595046125187, −6.70993427037605020851683552027, −5.29288132048050207955676289818, −3.95174373493091634596021260943, −1.71116836367403221573069884983,
1.71116836367403221573069884983, 3.95174373493091634596021260943, 5.29288132048050207955676289818, 6.70993427037605020851683552027, 8.247366710731016436595046125187, 8.788020066621735944813417079021, 10.74228228770574286361146919753, 11.06117180223364224220707158157, 12.42697619570053259749815883598, 13.53445154385831040337638587948
