| L(s) = 1 | + 2-s + 4-s + 8-s + 9-s − 12·13-s + 16-s − 4·17-s + 18-s − 12·26-s + 32-s − 4·34-s + 36-s − 4·37-s + 4·41-s − 10·49-s − 12·52-s − 12·53-s + 4·61-s + 64-s − 4·68-s + 72-s + 8·73-s − 4·74-s + 81-s + 4·82-s − 20·89-s + 16·97-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/3·9-s − 3.32·13-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 2.35·26-s + 0.176·32-s − 0.685·34-s + 1/6·36-s − 0.657·37-s + 0.624·41-s − 1.42·49-s − 1.66·52-s − 1.64·53-s + 0.512·61-s + 1/8·64-s − 0.485·68-s + 0.117·72-s + 0.936·73-s − 0.464·74-s + 1/9·81-s + 0.441·82-s − 2.11·89-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.040702798039804053791802466769, −8.342132676997866871617555456733, −7.72055975126333377666260774137, −7.47947512213883760565427163251, −6.87574139924512013673551975724, −6.64959962043719359186062300669, −5.91007962061405798709303677550, −5.13544146251726925083565635926, −4.89129507041850095151594897100, −4.50825772277642991881350186466, −3.79143046769365773246334925917, −2.89439976638041096085417523973, −2.45815773509881651106722995929, −1.78453255649523305607371273760, 0,
1.78453255649523305607371273760, 2.45815773509881651106722995929, 2.89439976638041096085417523973, 3.79143046769365773246334925917, 4.50825772277642991881350186466, 4.89129507041850095151594897100, 5.13544146251726925083565635926, 5.91007962061405798709303677550, 6.64959962043719359186062300669, 6.87574139924512013673551975724, 7.47947512213883760565427163251, 7.72055975126333377666260774137, 8.342132676997866871617555456733, 9.040702798039804053791802466769
