| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 7-s + 3·8-s + 9-s − 12-s − 6·13-s − 14-s − 16-s + 2·17-s − 18-s + 21-s − 6·23-s + 3·24-s − 5·25-s + 6·26-s + 27-s − 28-s + 4·31-s − 5·32-s − 2·34-s − 36-s − 6·37-s − 6·39-s + 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.288·12-s − 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.218·21-s − 1.25·23-s + 0.612·24-s − 25-s + 1.17·26-s + 0.192·27-s − 0.188·28-s + 0.718·31-s − 0.883·32-s − 0.342·34-s − 1/6·36-s − 0.986·37-s − 0.960·39-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.09934003052989, −15.66678194245057, −14.85494911030431, −14.39761827263310, −14.00808400989913, −13.54279244038615, −12.71236550201816, −12.28186668984802, −11.73521345923232, −10.88619088712078, −10.23687505995807, −9.808762316794452, −9.421871766472802, −8.785286616820768, −8.010837084342092, −7.766759569333846, −7.287608548009102, −6.370446379506326, −5.464441016301932, −4.914895450182639, −4.194467676476167, −3.684664968779810, −2.507994916270535, −2.047955055451924, −1.010936283001448, 0,
1.010936283001448, 2.047955055451924, 2.507994916270535, 3.684664968779810, 4.194467676476167, 4.914895450182639, 5.464441016301932, 6.370446379506326, 7.287608548009102, 7.766759569333846, 8.010837084342092, 8.785286616820768, 9.421871766472802, 9.808762316794452, 10.23687505995807, 10.88619088712078, 11.73521345923232, 12.28186668984802, 12.71236550201816, 13.54279244038615, 14.00808400989913, 14.39761827263310, 14.85494911030431, 15.66678194245057, 16.09934003052989
