| L(s) = 1 | − 2-s − 1.44·3-s + 4-s + 1.44·6-s − 3.25·7-s − 8-s − 0.918·9-s + 1.32·11-s − 1.44·12-s + 1.25·13-s + 3.25·14-s + 16-s − 4.62·17-s + 0.918·18-s − 3.37·19-s + 4.70·21-s − 1.32·22-s − 23-s + 1.44·24-s − 1.25·26-s + 5.65·27-s − 3.25·28-s + 4.29·29-s + 2.37·31-s − 32-s − 1.91·33-s + 4.62·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.832·3-s + 0.5·4-s + 0.588·6-s − 1.23·7-s − 0.353·8-s − 0.306·9-s + 0.400·11-s − 0.416·12-s + 0.349·13-s + 0.870·14-s + 0.250·16-s − 1.12·17-s + 0.216·18-s − 0.773·19-s + 1.02·21-s − 0.283·22-s − 0.208·23-s + 0.294·24-s − 0.246·26-s + 1.08·27-s − 0.615·28-s + 0.797·29-s + 0.426·31-s − 0.176·32-s − 0.333·33-s + 0.792·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5103907642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5103907642\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.44T + 3T^{2} \) |
| 7 | \( 1 + 3.25T + 7T^{2} \) |
| 11 | \( 1 - 1.32T + 11T^{2} \) |
| 13 | \( 1 - 1.25T + 13T^{2} \) |
| 17 | \( 1 + 4.62T + 17T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 - 5.74T + 37T^{2} \) |
| 41 | \( 1 + 5.13T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 6.06T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 2.72T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 6.60T + 67T^{2} \) |
| 71 | \( 1 - 0.265T + 71T^{2} \) |
| 73 | \( 1 + 5.26T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 2.02T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 8.62T + 97T^{2} \) |
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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
−9.983481667641971329497432300300, −8.870760191158605166972131246691, −8.482182548682995068442314885014, −7.07714723263211067758924349341, −6.42083595275768000363428050356, −5.98387518136225670465457428902, −4.69424964840080160838953101714, −3.48654432612481375745075590459, −2.32956423046191644517105556608, −0.59523310370207552611287165788,
0.59523310370207552611287165788, 2.32956423046191644517105556608, 3.48654432612481375745075590459, 4.69424964840080160838953101714, 5.98387518136225670465457428902, 6.42083595275768000363428050356, 7.07714723263211067758924349341, 8.482182548682995068442314885014, 8.870760191158605166972131246691, 9.983481667641971329497432300300
