L(s) = 1 | + 3-s − 3·4-s − 5-s + 9-s − 4·11-s − 3·12-s + 4·13-s − 15-s + 5·16-s + 3·20-s + 25-s + 27-s − 4·29-s − 4·33-s − 3·36-s + 4·39-s + 20·41-s − 8·43-s + 12·44-s − 45-s − 16·47-s + 5·48-s − 14·49-s − 12·52-s + 20·53-s + 4·55-s + 3·60-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 3/2·4-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.866·12-s + 1.10·13-s − 0.258·15-s + 5/4·16-s + 0.670·20-s + 1/5·25-s + 0.192·27-s − 0.742·29-s − 0.696·33-s − 1/2·36-s + 0.640·39-s + 3.12·41-s − 1.21·43-s + 1.80·44-s − 0.149·45-s − 2.33·47-s + 0.721·48-s − 2·49-s − 1.66·52-s + 2.74·53-s + 0.539·55-s + 0.387·60-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 408375 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 408375 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( 1 + T \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.327185665547056415560546606477, −8.168912727661556984599726575199, −7.68003844385765985083943135134, −7.22896894442775896645972178642, −6.58372794841850391610973143735, −5.86497653842965752491328535183, −5.57792290694465802264005507804, −4.87349408210187854122798421456, −4.52160444884874669934496061646, −4.01446647923640082173818964259, −3.45000960347210639177784403115, −3.01549784140686353642765162463, −2.13143823519478900872906310831, −1.11050393637334551379963274590, 0,
1.11050393637334551379963274590, 2.13143823519478900872906310831, 3.01549784140686353642765162463, 3.45000960347210639177784403115, 4.01446647923640082173818964259, 4.52160444884874669934496061646, 4.87349408210187854122798421456, 5.57792290694465802264005507804, 5.86497653842965752491328535183, 6.58372794841850391610973143735, 7.22896894442775896645972178642, 7.68003844385765985083943135134, 8.168912727661556984599726575199, 8.327185665547056415560546606477