| L(s) = 1 | − 2·5-s − 7-s − 11-s − 4·17-s − 4·19-s − 4·23-s − 25-s − 2·29-s + 2·31-s + 2·35-s − 6·37-s − 4·41-s + 4·43-s + 2·47-s + 49-s − 2·53-s + 2·55-s − 6·59-s + 4·61-s − 12·71-s + 16·73-s + 77-s + 8·79-s − 12·83-s + 8·85-s − 10·89-s + 8·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 0.301·11-s − 0.970·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 0.371·29-s + 0.359·31-s + 0.338·35-s − 0.986·37-s − 0.624·41-s + 0.609·43-s + 0.291·47-s + 1/7·49-s − 0.274·53-s + 0.269·55-s − 0.781·59-s + 0.512·61-s − 1.42·71-s + 1.87·73-s + 0.113·77-s + 0.900·79-s − 1.31·83-s + 0.867·85-s − 1.05·89-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5735399128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5735399128\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.30262701254731, −15.86716328445593, −15.41714141449524, −14.98686769432921, −14.18856244848622, −13.58265705266328, −13.08335972708972, −12.29146566437027, −12.07154469449418, −11.16092558507323, −10.84804777355415, −10.09416820242515, −9.480624551662315, −8.662493836080901, −8.282643264457376, −7.563976079680177, −6.934915352306940, −6.306634952300982, −5.601821071625448, −4.679339654217170, −4.106487319842271, −3.500591879925350, −2.566778784895122, −1.775522209513460, −0.3345148349639355,
0.3345148349639355, 1.775522209513460, 2.566778784895122, 3.500591879925350, 4.106487319842271, 4.679339654217170, 5.601821071625448, 6.306634952300982, 6.934915352306940, 7.563976079680177, 8.282643264457376, 8.662493836080901, 9.480624551662315, 10.09416820242515, 10.84804777355415, 11.16092558507323, 12.07154469449418, 12.29146566437027, 13.08335972708972, 13.58265705266328, 14.18856244848622, 14.98686769432921, 15.41714141449524, 15.86716328445593, 16.30262701254731
