L(s) = 1 | − 3-s + 1.82·5-s + 2.26·7-s + 9-s − 0.576·11-s − 1.28·13-s − 1.82·15-s − 1.88·17-s + 5.66·19-s − 2.26·21-s − 2.15·23-s − 1.66·25-s − 27-s + 6.70·29-s + 2.07·31-s + 0.576·33-s + 4.14·35-s − 5.63·37-s + 1.28·39-s + 8.23·41-s + 2.54·43-s + 1.82·45-s + 9.37·47-s − 1.84·49-s + 1.88·51-s + 6.17·53-s − 1.05·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.816·5-s + 0.857·7-s + 0.333·9-s − 0.173·11-s − 0.355·13-s − 0.471·15-s − 0.456·17-s + 1.29·19-s − 0.495·21-s − 0.449·23-s − 0.332·25-s − 0.192·27-s + 1.24·29-s + 0.372·31-s + 0.100·33-s + 0.700·35-s − 0.926·37-s + 0.205·39-s + 1.28·41-s + 0.388·43-s + 0.272·45-s + 1.36·47-s − 0.264·49-s + 0.263·51-s + 0.848·53-s − 0.141·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.066318131\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.066318131\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 - 1.82T + 5T^{2} \) |
| 7 | \( 1 - 2.26T + 7T^{2} \) |
| 11 | \( 1 + 0.576T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 + 1.88T + 17T^{2} \) |
| 19 | \( 1 - 5.66T + 19T^{2} \) |
| 23 | \( 1 + 2.15T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 - 2.07T + 31T^{2} \) |
| 37 | \( 1 + 5.63T + 37T^{2} \) |
| 41 | \( 1 - 8.23T + 41T^{2} \) |
| 43 | \( 1 - 2.54T + 43T^{2} \) |
| 47 | \( 1 - 9.37T + 47T^{2} \) |
| 53 | \( 1 - 6.17T + 53T^{2} \) |
| 59 | \( 1 + 3.25T + 59T^{2} \) |
| 61 | \( 1 - 1.95T + 61T^{2} \) |
| 67 | \( 1 - 8.37T + 67T^{2} \) |
| 71 | \( 1 + 2.61T + 71T^{2} \) |
| 73 | \( 1 + 5.82T + 73T^{2} \) |
| 79 | \( 1 + 1.10T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 3.01T + 89T^{2} \) |
| 97 | \( 1 + 5.58T + 97T^{2} \) |
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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
−8.409601431344482175728359499967, −7.65499208332553676217728433744, −6.97856674231412330959373726009, −6.08614299227285739444996703827, −5.47737127354476647533082364266, −4.85408030403706134163284435519, −4.04972419024414533560247706907, −2.76104468240840480031698080852, −1.90516489149977416127682917228, −0.880374650197562619114921898332,
0.880374650197562619114921898332, 1.90516489149977416127682917228, 2.76104468240840480031698080852, 4.04972419024414533560247706907, 4.85408030403706134163284435519, 5.47737127354476647533082364266, 6.08614299227285739444996703827, 6.97856674231412330959373726009, 7.65499208332553676217728433744, 8.409601431344482175728359499967