| L(s) = 1 | + 2-s − 4-s + 4·5-s − 3·8-s + 4·9-s + 4·10-s − 16-s + 4·18-s − 4·20-s − 12·23-s + 6·25-s + 4·31-s + 5·32-s − 4·36-s − 4·37-s − 12·40-s − 6·41-s + 12·43-s + 16·45-s − 12·46-s − 8·49-s + 6·50-s − 12·59-s + 4·61-s + 4·62-s + 7·64-s − 12·72-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.78·5-s − 1.06·8-s + 4/3·9-s + 1.26·10-s − 1/4·16-s + 0.942·18-s − 0.894·20-s − 2.50·23-s + 6/5·25-s + 0.718·31-s + 0.883·32-s − 2/3·36-s − 0.657·37-s − 1.89·40-s − 0.937·41-s + 1.82·43-s + 2.38·45-s − 1.76·46-s − 8/7·49-s + 0.848·50-s − 1.56·59-s + 0.512·61-s + 0.508·62-s + 7/8·64-s − 1.41·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.984109430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.984109430\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 41 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−10.40055180778932651559964881820, −9.911955960272971436354632936158, −9.693335626950667896037280526437, −9.325358158594172847696298594527, −8.496762909604681803744408490572, −7.993761647756516236575307062449, −7.21380793950831537304439496607, −6.36905598619971512541378504341, −6.13607935599421938681979021636, −5.54619282228675288982186307196, −4.85731632629106239954316493547, −4.25573246183196148686951291223, −3.58380881154477086499314784014, −2.41229588838619355243779657648, −1.64110821487634231113770540131,
1.64110821487634231113770540131, 2.41229588838619355243779657648, 3.58380881154477086499314784014, 4.25573246183196148686951291223, 4.85731632629106239954316493547, 5.54619282228675288982186307196, 6.13607935599421938681979021636, 6.36905598619971512541378504341, 7.21380793950831537304439496607, 7.993761647756516236575307062449, 8.496762909604681803744408490572, 9.325358158594172847696298594527, 9.693335626950667896037280526437, 9.911955960272971436354632936158, 10.40055180778932651559964881820
