| L(s) = 1 | + 2-s + 3-s + 4-s + 4.03·5-s + 6-s + 3.49·7-s + 8-s + 9-s + 4.03·10-s − 5.14·11-s + 12-s − 2.87·13-s + 3.49·14-s + 4.03·15-s + 16-s − 2.36·17-s + 18-s + 4.03·20-s + 3.49·21-s − 5.14·22-s + 4·23-s + 24-s + 11.2·25-s − 2.87·26-s + 27-s + 3.49·28-s + 1.43·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.80·5-s + 0.408·6-s + 1.32·7-s + 0.353·8-s + 0.333·9-s + 1.27·10-s − 1.55·11-s + 0.288·12-s − 0.797·13-s + 0.933·14-s + 1.04·15-s + 0.250·16-s − 0.572·17-s + 0.235·18-s + 0.901·20-s + 0.762·21-s − 1.09·22-s + 0.834·23-s + 0.204·24-s + 2.25·25-s − 0.563·26-s + 0.192·27-s + 0.660·28-s + 0.267·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.931834585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.931834585\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 4.03T + 5T^{2} \) |
| 7 | \( 1 - 3.49T + 7T^{2} \) |
| 11 | \( 1 + 5.14T + 11T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 + 2.36T + 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 1.43T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 + 8.29T + 37T^{2} \) |
| 41 | \( 1 - 6.69T + 41T^{2} \) |
| 43 | \( 1 + 4.83T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 2.67T + 59T^{2} \) |
| 61 | \( 1 - 5.12T + 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 - 0.831T + 71T^{2} \) |
| 73 | \( 1 + 7.78T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 0.223T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−9.089861973401178400769507083731, −8.275588603344922557990091695567, −7.46369233004518879548000566840, −6.70492816450603325349106882421, −5.54199579373797308991426975317, −5.18475746452626928948644186996, −4.49526449127165643949898377337, −2.89640984662650802481171939920, −2.31797760039170542883540589243, −1.57271934328142809640084157128,
1.57271934328142809640084157128, 2.31797760039170542883540589243, 2.89640984662650802481171939920, 4.49526449127165643949898377337, 5.18475746452626928948644186996, 5.54199579373797308991426975317, 6.70492816450603325349106882421, 7.46369233004518879548000566840, 8.275588603344922557990091695567, 9.089861973401178400769507083731
