| L(s) = 1 | + 2·3-s − 4·4-s − 2·5-s + 3·9-s − 8·12-s − 4·15-s + 12·16-s + 8·20-s − 12·23-s + 3·25-s + 4·27-s − 2·31-s − 12·36-s − 14·37-s − 6·45-s + 24·48-s − 13·49-s − 12·53-s − 24·59-s + 16·60-s − 32·64-s − 14·67-s − 24·69-s + 12·71-s + 6·75-s − 24·80-s + 5·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2·4-s − 0.894·5-s + 9-s − 2.30·12-s − 1.03·15-s + 3·16-s + 1.78·20-s − 2.50·23-s + 3/5·25-s + 0.769·27-s − 0.359·31-s − 2·36-s − 2.30·37-s − 0.894·45-s + 3.46·48-s − 1.85·49-s − 1.64·53-s − 3.12·59-s + 2.06·60-s − 4·64-s − 1.71·67-s − 2.88·69-s + 1.42·71-s + 0.692·75-s − 2.68·80-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−7.59654569582148090602817467428, −6.59523037473790869546358443506, −6.43362309054629922205991853367, −5.61750588322265071162144959368, −5.36747921289838822076634820154, −4.65725991292015173841856541050, −4.51845373560017717550608344934, −3.90838681993568032244631066955, −3.82028048843177336490572449531, −3.11214978990016222560190492173, −2.97589293278646371195021990890, −1.63448908436523374581135103796, −1.58360774879508418147674253120, 0, 0,
1.58360774879508418147674253120, 1.63448908436523374581135103796, 2.97589293278646371195021990890, 3.11214978990016222560190492173, 3.82028048843177336490572449531, 3.90838681993568032244631066955, 4.51845373560017717550608344934, 4.65725991292015173841856541050, 5.36747921289838822076634820154, 5.61750588322265071162144959368, 6.43362309054629922205991853367, 6.59523037473790869546358443506, 7.59654569582148090602817467428
