| L(s) = 1 | − 2·2-s − 4·3-s + 3·4-s + 8·6-s + 7-s − 4·8-s + 6·9-s − 12·12-s − 8·13-s − 2·14-s + 5·16-s + 12·17-s − 12·18-s − 4·21-s + 16·24-s − 10·25-s + 16·26-s + 4·27-s + 3·28-s − 4·31-s − 6·32-s − 24·34-s + 18·36-s + 32·39-s + 8·42-s − 20·48-s + 49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 3/2·4-s + 3.26·6-s + 0.377·7-s − 1.41·8-s + 2·9-s − 3.46·12-s − 2.21·13-s − 0.534·14-s + 5/4·16-s + 2.91·17-s − 2.82·18-s − 0.872·21-s + 3.26·24-s − 2·25-s + 3.13·26-s + 0.769·27-s + 0.566·28-s − 0.718·31-s − 1.06·32-s − 4.11·34-s + 3·36-s + 5.12·39-s + 1.23·42-s − 2.88·48-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1318492 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1318492 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| 31 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−7.57571100088867902110310233811, −7.53412797272985138081265407651, −6.94096468302848513947365341450, −6.50565556194215905004813007391, −5.87660226535060484767774489983, −5.57928681742950427486583645839, −5.42933804945618808612902258183, −4.97433533408618033814139302887, −4.31746354726812532740296695955, −3.51037205799184365501001984019, −2.86756904365859849040052255785, −2.18321351568779845969308070860, −1.40914032002296884552019261600, −0.67731023739110916602795338706, 0,
0.67731023739110916602795338706, 1.40914032002296884552019261600, 2.18321351568779845969308070860, 2.86756904365859849040052255785, 3.51037205799184365501001984019, 4.31746354726812532740296695955, 4.97433533408618033814139302887, 5.42933804945618808612902258183, 5.57928681742950427486583645839, 5.87660226535060484767774489983, 6.50565556194215905004813007391, 6.94096468302848513947365341450, 7.53412797272985138081265407651, 7.57571100088867902110310233811
