| L(s) = 1 | + 2·2-s + 2·4-s + 2·11-s − 13-s − 4·16-s + 19-s + 4·22-s − 2·26-s − 4·29-s + 9·31-s − 8·32-s − 3·37-s + 2·38-s + 10·41-s − 5·43-s + 4·44-s − 6·47-s − 2·52-s + 12·53-s − 8·58-s + 12·59-s + 10·61-s + 18·62-s − 8·64-s + 5·67-s + 6·71-s + 3·73-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.603·11-s − 0.277·13-s − 16-s + 0.229·19-s + 0.852·22-s − 0.392·26-s − 0.742·29-s + 1.61·31-s − 1.41·32-s − 0.493·37-s + 0.324·38-s + 1.56·41-s − 0.762·43-s + 0.603·44-s − 0.875·47-s − 0.277·52-s + 1.64·53-s − 1.05·58-s + 1.56·59-s + 1.28·61-s + 2.28·62-s − 64-s + 0.610·67-s + 0.712·71-s + 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.497073416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.497073416\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.30327974501316, −15.77088208367699, −15.13756043914085, −14.71863529710234, −14.16354828067862, −13.69992059616406, −13.08309521323749, −12.62446257657459, −11.94074474769368, −11.56189889623774, −11.01258758001279, −10.03764871100801, −9.611615035496698, −8.771850003791022, −8.216967618378850, −7.230814823229727, −6.749655276700428, −6.084227448637340, −5.427385161911774, −4.854436244587081, −4.090734731637058, −3.618627563033512, −2.748416788514357, −2.056879787365213, −0.7832319802456344,
0.7832319802456344, 2.056879787365213, 2.748416788514357, 3.618627563033512, 4.090734731637058, 4.854436244587081, 5.427385161911774, 6.084227448637340, 6.749655276700428, 7.230814823229727, 8.216967618378850, 8.771850003791022, 9.611615035496698, 10.03764871100801, 11.01258758001279, 11.56189889623774, 11.94074474769368, 12.62446257657459, 13.08309521323749, 13.69992059616406, 14.16354828067862, 14.71863529710234, 15.13756043914085, 15.77088208367699, 16.30327974501316
