| L(s) = 1 | − 4-s − 3·5-s + 7-s − 2·9-s + 16-s + 3·20-s + 3·25-s + 3·27-s − 28-s + 5·29-s − 31-s − 3·35-s + 2·36-s + 8·43-s + 6·45-s − 7·49-s + 9·61-s − 2·63-s − 64-s − 21·71-s − 22·73-s − 3·80-s + 4·81-s − 2·83-s − 24·97-s − 3·100-s − 3·108-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.34·5-s + 0.377·7-s − 2/3·9-s + 1/4·16-s + 0.670·20-s + 3/5·25-s + 0.577·27-s − 0.188·28-s + 0.928·29-s − 0.179·31-s − 0.507·35-s + 1/3·36-s + 1.21·43-s + 0.894·45-s − 49-s + 1.15·61-s − 0.251·63-s − 1/8·64-s − 2.49·71-s − 2.57·73-s − 0.335·80-s + 4/9·81-s − 0.219·83-s − 2.43·97-s − 0.299·100-s − 0.288·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142572 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142572 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 109 | $C_2$ | \( 1 + 10 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−8.993042615423311370155384516895, −8.414679621101458606274518275711, −8.256419159544516893060002068692, −7.74246600325439663006970362587, −7.18036076345162614855060965054, −6.78104734207297220940094130890, −5.90379585544922268278721992592, −5.61345263779580719514234390103, −4.71549949854265048599806214756, −4.47338439768838176066765034137, −3.86016531284218780925923522605, −3.16426129631516240359552882761, −2.59430579941224339112703045976, −1.27230187879966881746906090006, 0,
1.27230187879966881746906090006, 2.59430579941224339112703045976, 3.16426129631516240359552882761, 3.86016531284218780925923522605, 4.47338439768838176066765034137, 4.71549949854265048599806214756, 5.61345263779580719514234390103, 5.90379585544922268278721992592, 6.78104734207297220940094130890, 7.18036076345162614855060965054, 7.74246600325439663006970362587, 8.256419159544516893060002068692, 8.414679621101458606274518275711, 8.993042615423311370155384516895
