| L(s) = 1 | + 4-s + 2·7-s − 2·9-s − 8·13-s + 16-s − 10·25-s + 2·28-s − 6·29-s − 2·36-s + 3·49-s − 8·52-s + 12·53-s − 12·59-s − 4·63-s + 64-s − 8·67-s − 5·81-s − 12·83-s − 16·91-s − 10·100-s − 8·103-s + 24·107-s + 4·109-s + 2·112-s − 6·116-s + 16·117-s − 22·121-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s − 2/3·9-s − 2.21·13-s + 1/4·16-s − 2·25-s + 0.377·28-s − 1.11·29-s − 1/3·36-s + 3/7·49-s − 1.10·52-s + 1.64·53-s − 1.56·59-s − 0.503·63-s + 1/8·64-s − 0.977·67-s − 5/9·81-s − 1.31·83-s − 1.67·91-s − 100-s − 0.788·103-s + 2.32·107-s + 0.383·109-s + 0.188·112-s − 0.557·116-s + 1.47·117-s − 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164836 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164836 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 29 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.874952487363077123688486490575, −8.574673996165233070158070432652, −7.73899886874729111293892264969, −7.57571100088867902110310233811, −7.30118085295530868562669097184, −6.55315413557578547514079906546, −5.82169350283327601695140144520, −5.57928681742950427486583645839, −4.95666282602491657624262174992, −4.39532860206812970989814622572, −3.75399676073639417659089288658, −2.86651642125411249039840301372, −2.31158992149233556513632886331, −1.71520431274631175063040122688, 0,
1.71520431274631175063040122688, 2.31158992149233556513632886331, 2.86651642125411249039840301372, 3.75399676073639417659089288658, 4.39532860206812970989814622572, 4.95666282602491657624262174992, 5.57928681742950427486583645839, 5.82169350283327601695140144520, 6.55315413557578547514079906546, 7.30118085295530868562669097184, 7.57571100088867902110310233811, 7.73899886874729111293892264969, 8.574673996165233070158070432652, 8.874952487363077123688486490575
