L(s) = 1 | − 1.94·2-s − 1.49·3-s + 2.77·4-s + 5-s + 2.90·6-s + 1.13·7-s − 3.43·8-s + 1.24·9-s − 1.94·10-s − 0.709·11-s − 4.14·12-s + 13-s − 2.20·14-s − 1.49·15-s + 3.90·16-s + 1.77·17-s − 2.41·18-s + 2.77·20-s − 1.70·21-s + 1.37·22-s + 0.241·23-s + 5.14·24-s + 25-s − 1.94·26-s − 0.360·27-s + 3.14·28-s + 2.90·30-s + ⋯ |
L(s) = 1 | − 1.94·2-s − 1.49·3-s + 2.77·4-s + 5-s + 2.90·6-s + 1.13·7-s − 3.43·8-s + 1.24·9-s − 1.94·10-s − 0.709·11-s − 4.14·12-s + 13-s − 2.20·14-s − 1.49·15-s + 3.90·16-s + 1.77·17-s − 2.41·18-s + 2.77·20-s − 1.70·21-s + 1.37·22-s + 0.241·23-s + 5.14·24-s + 25-s − 1.94·26-s − 0.360·27-s + 3.14·28-s + 2.90·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4481774745\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4481774745\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + 1.94T + T^{2} \) |
| 3 | \( 1 + 1.49T + T^{2} \) |
| 7 | \( 1 - 1.13T + T^{2} \) |
| 11 | \( 1 + 0.709T + T^{2} \) |
| 17 | \( 1 - 1.77T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 0.241T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.94T + T^{2} \) |
| 47 | \( 1 - 0.241T + T^{2} \) |
| 53 | \( 1 + 0.709T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.709T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.13T + T^{2} \) |
| 97 | \( 1 + 1.49T + T^{2} \) |
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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
−9.558694572938584462409593386437, −8.450724256933496999661895032268, −8.019150457135728814287824353660, −7.06211980277848543889074864226, −6.27619253777148262097435373925, −5.66841373458882716898087155983, −5.01457607852384019771441982497, −3.01900977791589968834662152145, −1.66573551457843871328628517291, −1.05645648762942095662793994098,
1.05645648762942095662793994098, 1.66573551457843871328628517291, 3.01900977791589968834662152145, 5.01457607852384019771441982497, 5.66841373458882716898087155983, 6.27619253777148262097435373925, 7.06211980277848543889074864226, 8.019150457135728814287824353660, 8.450724256933496999661895032268, 9.558694572938584462409593386437