| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 2·7-s − 2·10-s − 11-s − 2·13-s + 4·14-s − 4·16-s − 2·17-s + 2·20-s + 2·22-s + 4·23-s + 25-s + 4·26-s − 4·28-s − 5·29-s − 9·31-s + 8·32-s + 4·34-s − 2·35-s + 6·37-s + 6·41-s − 10·43-s − 2·44-s − 8·46-s − 3·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 0.755·7-s − 0.632·10-s − 0.301·11-s − 0.554·13-s + 1.06·14-s − 16-s − 0.485·17-s + 0.447·20-s + 0.426·22-s + 0.834·23-s + 1/5·25-s + 0.784·26-s − 0.755·28-s − 0.928·29-s − 1.61·31-s + 1.41·32-s + 0.685·34-s − 0.338·35-s + 0.986·37-s + 0.937·41-s − 1.52·43-s − 0.301·44-s − 1.17·46-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4492485878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4492485878\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−16.18225623696845, −15.64318284683644, −14.82185059813068, −14.56370504848972, −13.45371440170632, −13.21924924958263, −12.71004364014136, −11.87835266797893, −11.12733761060887, −10.79015202988201, −10.13799177428838, −9.519884976881647, −9.271232335613251, −8.729982058443670, −7.879603632123751, −7.441312084124513, −6.807158897839503, −6.263690779806886, −5.403147159586525, −4.763127068157815, −3.829209172547316, −2.928629259318299, −2.207368834790888, −1.463180695101971, −0.3632343960846808,
0.3632343960846808, 1.463180695101971, 2.207368834790888, 2.928629259318299, 3.829209172547316, 4.763127068157815, 5.403147159586525, 6.263690779806886, 6.807158897839503, 7.441312084124513, 7.879603632123751, 8.729982058443670, 9.271232335613251, 9.519884976881647, 10.13799177428838, 10.79015202988201, 11.12733761060887, 11.87835266797893, 12.71004364014136, 13.21924924958263, 13.45371440170632, 14.56370504848972, 14.82185059813068, 15.64318284683644, 16.18225623696845
