| L(s) = 1 | − 2-s + 4-s + 1.87·5-s + 0.347·7-s − 8-s − 1.87·10-s − 1.53·11-s − 0.347·14-s + 16-s + 1.87·20-s + 1.53·22-s + 2.53·25-s + 0.347·28-s + 29-s + 1.53·31-s − 32-s + 0.652·35-s − 1.87·40-s − 1.53·44-s − 0.879·49-s − 2.53·50-s − 0.347·53-s − 2.87·55-s − 0.347·56-s − 58-s + 59-s − 1.53·62-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 1.87·5-s + 0.347·7-s − 8-s − 1.87·10-s − 1.53·11-s − 0.347·14-s + 16-s + 1.87·20-s + 1.53·22-s + 2.53·25-s + 0.347·28-s + 29-s + 1.53·31-s − 32-s + 0.652·35-s − 1.87·40-s − 1.53·44-s − 0.879·49-s − 2.53·50-s − 0.347·53-s − 2.87·55-s − 0.347·56-s − 58-s + 59-s − 1.53·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.036270604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.036270604\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 1.87T + T^{2} \) |
| 7 | \( 1 - 0.347T + T^{2} \) |
| 11 | \( 1 + 1.53T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 - 1.53T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 0.347T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.347T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + 0.347T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.87T + T^{2} \) |
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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
−9.575265339835743534307026387224, −8.567829111662892568415076272962, −8.093605303178597732526453541498, −7.01629530945912672160229337612, −6.28154377639610931461103043614, −5.55825936805979295729179718378, −4.84061598291919795815450810712, −2.87859354304462526185304447061, −2.36727086647577685208815237210, −1.29431574017196265416246133313,
1.29431574017196265416246133313, 2.36727086647577685208815237210, 2.87859354304462526185304447061, 4.84061598291919795815450810712, 5.55825936805979295729179718378, 6.28154377639610931461103043614, 7.01629530945912672160229337612, 8.093605303178597732526453541498, 8.567829111662892568415076272962, 9.575265339835743534307026387224
