| L(s) = 1 | + 2-s + 4-s + 0.347·5-s + 1.53·7-s + 8-s + 0.347·10-s − 1.87·11-s + 1.53·14-s + 16-s + 0.347·20-s − 1.87·22-s − 0.879·25-s + 1.53·28-s − 29-s − 1.87·31-s + 32-s + 0.532·35-s + 0.347·40-s − 1.87·44-s + 1.34·49-s − 0.879·50-s + 1.53·53-s − 0.652·55-s + 1.53·56-s − 58-s − 59-s − 1.87·62-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 0.347·5-s + 1.53·7-s + 8-s + 0.347·10-s − 1.87·11-s + 1.53·14-s + 16-s + 0.347·20-s − 1.87·22-s − 0.879·25-s + 1.53·28-s − 29-s − 1.87·31-s + 32-s + 0.532·35-s + 0.347·40-s − 1.87·44-s + 1.34·49-s − 0.879·50-s + 1.53·53-s − 0.652·55-s + 1.53·56-s − 58-s − 59-s − 1.87·62-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.434469339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.434469339\) |
| \(L(1)\) |
|
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 | \( 1 - 0.347T + T^{2} \) |
| 7 | \( 1 - 1.53T + T^{2} \) |
| 11 | \( 1 + 1.87T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + 1.87T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.53T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.53T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - 1.53T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 0.347T + T^{2} \) |
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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
−9.460926325611990237931164196594, −8.248657873933707644746410608189, −7.73345599899026477150537858469, −7.12055370706312490695017985342, −5.69889848288708428412709476834, −5.42263622262079504387285885681, −4.66013257151085150313069505521, −3.66227289075139448412540855009, −2.41861866997818736355041107047, −1.78267083195715027470346593981,
1.78267083195715027470346593981, 2.41861866997818736355041107047, 3.66227289075139448412540855009, 4.66013257151085150313069505521, 5.42263622262079504387285885681, 5.69889848288708428412709476834, 7.12055370706312490695017985342, 7.73345599899026477150537858469, 8.248657873933707644746410608189, 9.460926325611990237931164196594
