| L(s) = 1 | + 2·2-s − 5·4-s − 20·8-s − 24·9-s + 5·16-s − 48·18-s − 84·25-s − 40·29-s + 118·32-s + 120·36-s + 216·37-s − 168·50-s − 120·53-s − 80·58-s + 111·64-s + 480·72-s + 432·74-s + 270·81-s + 420·100-s − 240·106-s − 600·109-s + 720·113-s + 200·116-s + 364·121-s + 127-s − 480·128-s + 131-s + ⋯ |
| L(s) = 1 | + 2-s − 5/4·4-s − 5/2·8-s − 8/3·9-s + 5/16·16-s − 8/3·18-s − 3.35·25-s − 1.37·29-s + 3.68·32-s + 10/3·36-s + 5.83·37-s − 3.35·50-s − 2.26·53-s − 1.37·58-s + 1.73·64-s + 20/3·72-s + 5.83·74-s + 10/3·81-s + 21/5·100-s − 2.26·106-s − 5.50·109-s + 6.37·113-s + 1.72·116-s + 3.00·121-s + 0.00787·127-s − 3.75·128-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4088742751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4088742751\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 4 p T^{2} + p^{4} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 42 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 182 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 210 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 240 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 692 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 818 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 842 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 54 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 1680 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3638 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 1418 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 6692 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 5130 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 8018 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 1678 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 9600 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 6958 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 13508 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 4560 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 16080 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.996677221061343074943449868454, −8.619707172970680178244451293059, −8.281642112681730993521714664768, −8.110902313928435848885388436779, −7.944281787903386255965148194792, −7.71974672073860357028587689810, −7.52091289358258886060711181164, −6.89313132538762087742290967353, −6.24848875534064756405206227930, −6.20297670192299774598135329692, −5.89276526988162146194851463057, −5.83819061856799236660009809320, −5.59004644697935647767649616655, −5.36808146775301525689972057420, −4.72762913355675682061767706516, −4.41465462998141866202743900233, −4.38546323280282812037056861326, −3.94620010463490645273883208167, −3.37945801024732984478611045958, −3.32572444513599779327041976756, −2.85575017581236808969214129484, −2.43375627917662648988963881580, −2.04138010205948689785267214969, −0.878390236975194931876675959308, −0.21090593768300211095272762186,
0.21090593768300211095272762186, 0.878390236975194931876675959308, 2.04138010205948689785267214969, 2.43375627917662648988963881580, 2.85575017581236808969214129484, 3.32572444513599779327041976756, 3.37945801024732984478611045958, 3.94620010463490645273883208167, 4.38546323280282812037056861326, 4.41465462998141866202743900233, 4.72762913355675682061767706516, 5.36808146775301525689972057420, 5.59004644697935647767649616655, 5.83819061856799236660009809320, 5.89276526988162146194851463057, 6.20297670192299774598135329692, 6.24848875534064756405206227930, 6.89313132538762087742290967353, 7.52091289358258886060711181164, 7.71974672073860357028587689810, 7.944281787903386255965148194792, 8.110902313928435848885388436779, 8.281642112681730993521714664768, 8.619707172970680178244451293059, 8.996677221061343074943449868454
