| L(s) = 1 | + 2-s + 4-s − 4·5-s + 8-s − 4·10-s + 4·11-s − 13-s + 16-s + 5·17-s + 7·19-s − 4·20-s + 4·22-s − 23-s + 11·25-s − 26-s − 5·29-s + 4·31-s + 32-s + 5·34-s + 37-s + 7·38-s − 4·40-s − 10·41-s − 10·43-s + 4·44-s − 46-s − 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.353·8-s − 1.26·10-s + 1.20·11-s − 0.277·13-s + 1/4·16-s + 1.21·17-s + 1.60·19-s − 0.894·20-s + 0.852·22-s − 0.208·23-s + 11/5·25-s − 0.196·26-s − 0.928·29-s + 0.718·31-s + 0.176·32-s + 0.857·34-s + 0.164·37-s + 1.13·38-s − 0.632·40-s − 1.56·41-s − 1.52·43-s + 0.603·44-s − 0.147·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46746 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46746 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−14.86655675142776, −14.43737391477209, −13.89727732273926, −13.38708272607615, −12.61450319369008, −12.04403627124423, −11.93084611971274, −11.46204456615868, −11.07645324494302, −10.10469576079704, −9.813093911198658, −9.011408343532041, −8.378111563896036, −7.812680708881092, −7.461457262161027, −6.884044515743156, −6.382530623049129, −5.503445904491975, −4.992877562943336, −4.413853893056945, −3.732472369607414, −3.360348028447731, −2.971393358657012, −1.639710086905014, −1.056990429111445, 0,
1.056990429111445, 1.639710086905014, 2.971393358657012, 3.360348028447731, 3.732472369607414, 4.413853893056945, 4.992877562943336, 5.503445904491975, 6.382530623049129, 6.884044515743156, 7.461457262161027, 7.812680708881092, 8.378111563896036, 9.011408343532041, 9.813093911198658, 10.10469576079704, 11.07645324494302, 11.46204456615868, 11.93084611971274, 12.04403627124423, 12.61450319369008, 13.38708272607615, 13.89727732273926, 14.43737391477209, 14.86655675142776
