L(s) = 1 | − 3-s − 5-s + 9-s + 2·13-s + 15-s + 6·17-s + 4·23-s + 25-s − 27-s − 2·29-s − 8·31-s + 6·37-s − 2·39-s + 6·41-s − 12·43-s − 45-s − 12·47-s − 6·51-s − 10·53-s + 8·59-s + 10·61-s − 2·65-s + 12·67-s − 4·69-s − 8·71-s − 10·73-s − 75-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 1.45·17-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 1.82·43-s − 0.149·45-s − 1.75·47-s − 0.840·51-s − 1.37·53-s + 1.04·59-s + 1.28·61-s − 0.248·65-s + 1.46·67-s − 0.481·69-s − 0.949·71-s − 1.17·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.97786003328105, −15.02587210560577, −14.72477618637465, −14.30223192546002, −13.36715145936599, −12.87773533726518, −12.63079700957966, −11.70765988050576, −11.43128164607868, −10.98356588496325, −10.23943839172553, −9.747559326825511, −9.178361889426183, −8.372228374367733, −7.923219088837230, −7.280895926248607, −6.718678277382517, −6.048723603666450, −5.375397792742175, −4.978535635228911, −4.059764597701272, −3.511250323463933, −2.873864354844476, −1.695672871307937, −1.043234091926824, 0,
1.043234091926824, 1.695672871307937, 2.873864354844476, 3.511250323463933, 4.059764597701272, 4.978535635228911, 5.375397792742175, 6.048723603666450, 6.718678277382517, 7.280895926248607, 7.923219088837230, 8.372228374367733, 9.178361889426183, 9.747559326825511, 10.23943839172553, 10.98356588496325, 11.43128164607868, 11.70765988050576, 12.63079700957966, 12.87773533726518, 13.36715145936599, 14.30223192546002, 14.72477618637465, 15.02587210560577, 15.97786003328105