L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 12-s − 13-s + 16-s − 17-s + 19-s + 23-s − 24-s + 25-s + 26-s − 27-s − 29-s − 2·31-s − 32-s + 34-s − 37-s − 38-s − 39-s − 46-s + 48-s + 49-s − 50-s − 51-s + ⋯ |
L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 12-s − 13-s + 16-s − 17-s + 19-s + 23-s − 24-s + 25-s + 26-s − 27-s − 29-s − 2·31-s − 32-s + 34-s − 37-s − 38-s − 39-s − 46-s + 48-s + 49-s − 50-s − 51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6016271087\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6016271087\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 53 | \( 1 - T \) |
good | 3 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 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 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−12.50104207939304209104938510389, −11.39967471631552014274489218820, −10.50186667170449994165901255567, −9.154088438944714961936760612840, −9.015000219394991271234678530123, −7.64907334828837717882700327518, −7.00892181179622288372976982618, −5.36268031244416550716130776826, −3.36221873509857810706285143775, −2.18792524840052695630130637294,
2.18792524840052695630130637294, 3.36221873509857810706285143775, 5.36268031244416550716130776826, 7.00892181179622288372976982618, 7.64907334828837717882700327518, 9.015000219394991271234678530123, 9.154088438944714961936760612840, 10.50186667170449994165901255567, 11.39967471631552014274489218820, 12.50104207939304209104938510389