| L(s) = 1 | − 2-s − 5-s − 7-s + 8-s + 9-s + 10-s + 14-s − 16-s − 18-s − 19-s + 31-s + 35-s + 38-s − 40-s − 41-s − 45-s + 2·47-s − 56-s − 59-s − 62-s − 63-s + 64-s + 2·67-s − 70-s − 71-s + 72-s + 80-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s − 7-s + 8-s + 9-s + 10-s + 14-s − 16-s − 18-s − 19-s + 31-s + 35-s + 38-s − 40-s − 41-s − 45-s + 2·47-s − 56-s − 59-s − 62-s − 63-s + 64-s + 2·67-s − 70-s − 71-s + 72-s + 80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2177494736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2177494736\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−17.30044677342347514916214466508, −16.18729920448603777480556373415, −15.36955016827026145573435350782, −13.47643814207973877644675198519, −12.31888813921751727125231993793, −10.63965438412188612144155604195, −9.568262511391337160706292941451, −8.198165198685797618624379681819, −6.92572000323394459470037275402, −4.16621475268391289462455833400,
4.16621475268391289462455833400, 6.92572000323394459470037275402, 8.198165198685797618624379681819, 9.568262511391337160706292941451, 10.63965438412188612144155604195, 12.31888813921751727125231993793, 13.47643814207973877644675198519, 15.36955016827026145573435350782, 16.18729920448603777480556373415, 17.30044677342347514916214466508
