| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 4·11-s + 12-s + 2·13-s + 14-s − 15-s + 16-s − 2·17-s − 18-s − 4·19-s − 20-s − 21-s + 4·22-s + 8·23-s − 24-s + 25-s − 2·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.852·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 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
−13.15665985605770, −12.65602417226229, −12.45484535346406, −11.71209940271277, −11.06209296497161, −10.94400706333372, −10.34436108561798, −10.09872826886554, −9.316623155223501, −8.878129293496139, −8.749819207454602, −8.059567461758888, −7.710923419571995, −7.267122879362523, −6.735261913663520, −6.260141361457757, −5.773040632859250, −4.934225697295593, −4.643540793682552, −3.975188686092242, −3.277428597651877, −2.816290771488825, −2.579652972806318, −1.582947039819907, −1.242040132653579, 0, 0,
1.242040132653579, 1.582947039819907, 2.579652972806318, 2.816290771488825, 3.277428597651877, 3.975188686092242, 4.643540793682552, 4.934225697295593, 5.773040632859250, 6.260141361457757, 6.735261913663520, 7.267122879362523, 7.710923419571995, 8.059567461758888, 8.749819207454602, 8.878129293496139, 9.316623155223501, 10.09872826886554, 10.34436108561798, 10.94400706333372, 11.06209296497161, 11.71209940271277, 12.45484535346406, 12.65602417226229, 13.15665985605770
