| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s + 13-s + 15-s + 16-s + 17-s + 18-s + 19-s − 20-s − 24-s + 26-s − 27-s + 30-s − 31-s + 32-s + 34-s + 36-s + 38-s − 39-s − 40-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s + 13-s + 15-s + 16-s + 17-s + 18-s + 19-s − 20-s − 24-s + 26-s − 27-s + 30-s − 31-s + 32-s + 34-s + 36-s + 38-s − 39-s − 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.307696115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.307696115\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 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
−10.57234385888703938530000410016, −9.653729732998442450855909727690, −8.159566801347177302612455568649, −7.46261537196194152962088416134, −6.65927594779266654447167594717, −5.72114314160017851274084188763, −5.04832573581779530329145162586, −3.96416847856150163924792326724, −3.33760011976684382992997045715, −1.41557719550640901446967341341,
1.41557719550640901446967341341, 3.33760011976684382992997045715, 3.96416847856150163924792326724, 5.04832573581779530329145162586, 5.72114314160017851274084188763, 6.65927594779266654447167594717, 7.46261537196194152962088416134, 8.159566801347177302612455568649, 9.653729732998442450855909727690, 10.57234385888703938530000410016
