| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 11-s + 12-s − 15-s + 16-s + 18-s − 20-s − 22-s + 24-s + 27-s − 29-s − 30-s − 31-s + 32-s − 33-s + 36-s − 40-s − 44-s − 45-s + 48-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 11-s + 12-s − 15-s + 16-s + 18-s − 20-s − 22-s + 24-s + 27-s − 29-s − 30-s − 31-s + 32-s − 33-s + 36-s − 40-s − 44-s − 45-s + 48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.230679896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.230679896\) |
| \(L(1)\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.09099054614147659899985838218, −9.056911787024093838332940118845, −7.911783537969590876281261297802, −7.68646073319634890662737941471, −6.78607974237754796872369371950, −5.55419437561912647388496781785, −4.58769920616123614729348756471, −3.76162732913248761880868551009, −3.03553903182846132016218102561, −1.93285963023290371336841907255,
1.93285963023290371336841907255, 3.03553903182846132016218102561, 3.76162732913248761880868551009, 4.58769920616123614729348756471, 5.55419437561912647388496781785, 6.78607974237754796872369371950, 7.68646073319634890662737941471, 7.911783537969590876281261297802, 9.056911787024093838332940118845, 10.09099054614147659899985838218
