| L(s) = 1 | − 2-s + 4-s − 8-s − 3·11-s + 13-s + 16-s + 6·17-s + 2·19-s + 3·22-s − 26-s − 3·29-s − 4·31-s − 32-s − 6·34-s + 2·37-s − 2·38-s + 3·41-s − 43-s − 3·44-s + 6·47-s + 52-s − 9·53-s + 3·58-s + 9·59-s − 4·61-s + 4·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.904·11-s + 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.458·19-s + 0.639·22-s − 0.196·26-s − 0.557·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.324·38-s + 0.468·41-s − 0.152·43-s − 0.452·44-s + 0.875·47-s + 0.138·52-s − 1.23·53-s + 0.393·58-s + 1.17·59-s − 0.512·61-s + 0.508·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.041341175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.041341175\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−12.75092404163511, −12.24889595439815, −11.72874903495727, −11.27167816885638, −10.83596948778127, −10.38946162165859, −9.938543176832636, −9.578658977879202, −9.013340889577099, −8.633490145546816, −7.928342058830129, −7.631596619820038, −7.429699015993905, −6.676135551239491, −6.087619190442224, −5.691168150092748, −5.200279812405636, −4.723892653229400, −3.869999972976466, −3.384518866616449, −2.980906309916614, −2.210119480527538, −1.784290836552769, −0.9308310743826735, −0.5217712169808293,
0.5217712169808293, 0.9308310743826735, 1.784290836552769, 2.210119480527538, 2.980906309916614, 3.384518866616449, 3.869999972976466, 4.723892653229400, 5.200279812405636, 5.691168150092748, 6.087619190442224, 6.676135551239491, 7.429699015993905, 7.631596619820038, 7.928342058830129, 8.633490145546816, 9.013340889577099, 9.578658977879202, 9.938543176832636, 10.38946162165859, 10.83596948778127, 11.27167816885638, 11.72874903495727, 12.24889595439815, 12.75092404163511
