L(s) = 1 | + 4-s − 5-s + 9-s − 11-s − 13-s + 16-s − 19-s − 20-s + 36-s − 37-s − 43-s − 44-s − 45-s − 47-s + 49-s − 52-s + 2·53-s + 55-s + 2·59-s + 64-s + 65-s − 71-s + 2·73-s − 76-s − 79-s − 80-s + 81-s + ⋯ |
L(s) = 1 | + 4-s − 5-s + 9-s − 11-s − 13-s + 16-s − 19-s − 20-s + 36-s − 37-s − 43-s − 44-s − 45-s − 47-s + 49-s − 52-s + 2·53-s + 55-s + 2·59-s + 64-s + 65-s − 71-s + 2·73-s − 76-s − 79-s − 80-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7358031812\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7358031812\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 211 | \( 1 - T \) |
good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−12.39814787958180065980360115033, −11.74104219053413987752461018312, −10.64078657438421046185276150704, −10.01038163356747423234171091982, −8.324932616794407432977756050213, −7.46386740565527729315092106349, −6.77516729846439138358216471580, −5.18037386937466737132796711185, −3.83733924492078633030089718140, −2.29906709754737556376540443181,
2.29906709754737556376540443181, 3.83733924492078633030089718140, 5.18037386937466737132796711185, 6.77516729846439138358216471580, 7.46386740565527729315092106349, 8.324932616794407432977756050213, 10.01038163356747423234171091982, 10.64078657438421046185276150704, 11.74104219053413987752461018312, 12.39814787958180065980360115033