| L(s) = 1 | − 5-s − 6·11-s − 17-s + 3·19-s + 7·23-s − 4·25-s − 6·29-s + 8·31-s + 7·37-s + 9·43-s − 6·47-s − 2·53-s + 6·55-s − 11·59-s − 6·61-s − 9·67-s − 9·71-s − 12·73-s + 8·79-s + 4·83-s + 85-s − 5·89-s − 3·95-s − 4·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.80·11-s − 0.242·17-s + 0.688·19-s + 1.45·23-s − 4/5·25-s − 1.11·29-s + 1.43·31-s + 1.15·37-s + 1.37·43-s − 0.875·47-s − 0.274·53-s + 0.809·55-s − 1.43·59-s − 0.768·61-s − 1.09·67-s − 1.06·71-s − 1.40·73-s + 0.900·79-s + 0.439·83-s + 0.108·85-s − 0.529·89-s − 0.307·95-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.69255701909404, −13.22775444937341, −12.96293793638575, −12.45396880262638, −11.75108106340454, −11.42604477855795, −10.90377799436877, −10.44330044280368, −10.02676384447803, −9.265240705651378, −9.080343506583137, −8.218446682339739, −7.785995646248282, −7.542150594645278, −7.026058065430539, −6.122930675728388, −5.841904846759614, −5.150951654265394, −4.643644551141169, −4.269396509263675, −3.257377070123255, −2.986563930643171, −2.401790017830965, −1.579125577851934, −0.7310804438124568, 0,
0.7310804438124568, 1.579125577851934, 2.401790017830965, 2.986563930643171, 3.257377070123255, 4.269396509263675, 4.643644551141169, 5.150951654265394, 5.841904846759614, 6.122930675728388, 7.026058065430539, 7.542150594645278, 7.785995646248282, 8.218446682339739, 9.080343506583137, 9.265240705651378, 10.02676384447803, 10.44330044280368, 10.90377799436877, 11.42604477855795, 11.75108106340454, 12.45396880262638, 12.96293793638575, 13.22775444937341, 13.69255701909404
