| L(s) = 1 | + 2·2-s − 4-s − 5-s − 8·8-s + 9-s − 2·10-s + 2·13-s − 7·16-s + 2·18-s + 20-s + 25-s + 4·26-s − 4·29-s + 14·32-s − 36-s + 20·37-s + 8·40-s − 45-s − 16·47-s − 14·49-s + 2·50-s − 2·52-s − 8·58-s − 4·61-s + 35·64-s − 2·65-s − 24·67-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 0.447·5-s − 2.82·8-s + 1/3·9-s − 0.632·10-s + 0.554·13-s − 7/4·16-s + 0.471·18-s + 0.223·20-s + 1/5·25-s + 0.784·26-s − 0.742·29-s + 2.47·32-s − 1/6·36-s + 3.28·37-s + 1.26·40-s − 0.149·45-s − 2.33·47-s − 2·49-s + 0.282·50-s − 0.277·52-s − 1.05·58-s − 0.512·61-s + 35/8·64-s − 0.248·65-s − 2.93·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( 1 + T \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.843956595114680827452383410006, −8.526905706632633062975018882770, −7.76910750368042878757814163764, −7.68003844385765985083943135134, −6.63078439380576833175908493602, −6.20481649249773216903893847569, −5.86497653842965752491328535183, −5.27152076797277019902163228084, −4.52160444884874669934496061646, −4.46342990915686607443282092868, −3.92265287197659703286168465477, −3.01746980794826050429743741143, −3.01549784140686353642765162463, −1.41836019676092930181617969749, 0,
1.41836019676092930181617969749, 3.01549784140686353642765162463, 3.01746980794826050429743741143, 3.92265287197659703286168465477, 4.46342990915686607443282092868, 4.52160444884874669934496061646, 5.27152076797277019902163228084, 5.86497653842965752491328535183, 6.20481649249773216903893847569, 6.63078439380576833175908493602, 7.68003844385765985083943135134, 7.76910750368042878757814163764, 8.526905706632633062975018882770, 8.843956595114680827452383410006
