| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 13-s + 14-s + 16-s + 17-s + 20-s − 26-s − 28-s + 2·31-s − 32-s − 34-s − 35-s − 37-s − 40-s − 43-s + 47-s + 52-s + 56-s − 2·62-s + 64-s + 65-s + 68-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 13-s + 14-s + 16-s + 17-s + 20-s − 26-s − 28-s + 2·31-s − 32-s − 34-s − 35-s − 37-s − 40-s − 43-s + 47-s + 52-s + 56-s − 2·62-s + 64-s + 65-s + 68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7493759327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7493759327\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−10.04538447887338157989606832290, −9.590168661714193585167030455251, −8.718567837721073772849137194548, −7.930635037911471552829375101604, −6.73503320946955415370360755074, −6.23192187633918577809210803507, −5.41499807564931199081157502266, −3.60752461315762887217628336411, −2.64096071576240479043933942638, −1.31290686729774965324606450596,
1.31290686729774965324606450596, 2.64096071576240479043933942638, 3.60752461315762887217628336411, 5.41499807564931199081157502266, 6.23192187633918577809210803507, 6.73503320946955415370360755074, 7.930635037911471552829375101604, 8.718567837721073772849137194548, 9.590168661714193585167030455251, 10.04538447887338157989606832290
