| L(s) = 1 | − 2-s − 4-s − 2·5-s + 7-s + 3·8-s + 2·10-s + 11-s − 2·13-s − 14-s − 16-s − 2·17-s + 2·20-s − 22-s + 8·23-s − 25-s + 2·26-s − 28-s − 10·29-s + 8·31-s − 5·32-s + 2·34-s − 2·35-s + 2·37-s − 6·40-s − 2·41-s − 12·43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s + 1.06·8-s + 0.632·10-s + 0.301·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.447·20-s − 0.213·22-s + 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s − 1.85·29-s + 1.43·31-s − 0.883·32-s + 0.342·34-s − 0.338·35-s + 0.328·37-s − 0.948·40-s − 0.312·41-s − 1.82·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.23837742175291, −12.51464922105609, −12.10886605810044, −11.63656404431875, −11.08860463611148, −10.88263650271584, −10.27978185098864, −9.657508058408825, −9.387775996383438, −8.852917809916588, −8.425786624832735, −8.001103666182721, −7.528745884820063, −7.121510493762901, −6.702456726412829, −5.917823487672663, −5.254681597055378, −4.807109176062579, −4.463893195791705, −3.783271024572412, −3.456141719478227, −2.610262600204602, −1.965672829641043, −1.274189421231413, −0.6519808065857946, 0,
0.6519808065857946, 1.274189421231413, 1.965672829641043, 2.610262600204602, 3.456141719478227, 3.783271024572412, 4.463893195791705, 4.807109176062579, 5.254681597055378, 5.917823487672663, 6.702456726412829, 7.121510493762901, 7.528745884820063, 8.001103666182721, 8.425786624832735, 8.852917809916588, 9.387775996383438, 9.657508058408825, 10.27978185098864, 10.88263650271584, 11.08860463611148, 11.63656404431875, 12.10886605810044, 12.51464922105609, 13.23837742175291
