L(s) = 1 | + 3-s + 5-s + 9-s − 4·11-s − 6·13-s + 15-s − 6·17-s + 4·19-s + 4·23-s + 25-s + 27-s + 2·29-s + 8·31-s − 4·33-s − 6·37-s − 6·39-s − 6·41-s + 8·43-s + 45-s − 6·51-s − 6·53-s − 4·55-s + 4·57-s + 4·59-s + 10·61-s − 6·65-s + 8·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s − 0.960·39-s − 0.937·41-s + 1.21·43-s + 0.149·45-s − 0.840·51-s − 0.824·53-s − 0.539·55-s + 0.529·57-s + 0.520·59-s + 1.28·61-s − 0.744·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.95951113229490, −14.16323180805233, −13.86367948174847, −13.40496354070258, −12.80537693141522, −12.45682847776855, −11.80888864112788, −11.19149828076371, −10.55482748273469, −10.12860170159857, −9.621962169193698, −9.174173232543254, −8.521322329945996, −7.996616700515775, −7.458168231882954, −6.854406126017366, −6.509325386933500, −5.454458930660638, −4.999851721235811, −4.703831582985371, −3.784391692469373, −2.913964075270191, −2.516899307049657, −2.077594707619333, −0.9722016419150248, 0,
0.9722016419150248, 2.077594707619333, 2.516899307049657, 2.913964075270191, 3.784391692469373, 4.703831582985371, 4.999851721235811, 5.454458930660638, 6.509325386933500, 6.854406126017366, 7.458168231882954, 7.996616700515775, 8.521322329945996, 9.174173232543254, 9.621962169193698, 10.12860170159857, 10.55482748273469, 11.19149828076371, 11.80888864112788, 12.45682847776855, 12.80537693141522, 13.40496354070258, 13.86367948174847, 14.16323180805233, 14.95951113229490