| L(s) = 1 | − 8·13-s + 4·25-s − 2·49-s − 56·61-s − 24·73-s − 56·97-s + 48·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 2.21·13-s + 4/5·25-s − 2/7·49-s − 7.17·61-s − 2.80·73-s − 5.68·97-s + 4.59·109-s − 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6486425830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6486425830\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 88 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
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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
−6.45395562642449660141746778741, −6.43441876828900572970375733728, −6.25456939991234159985501383387, −5.70958883213699101794291930542, −5.64739558181282096177381646039, −5.45134145812226616470449838336, −5.37244848859719355475287577929, −4.83705498833167358416491705276, −4.74802571157142289038999913584, −4.61929811004212363961206708831, −4.46101054022409817401880442905, −4.22470355973473536538549117117, −4.00924234978420940382886947876, −3.59958093013743889585048963707, −3.32725408469917901749656852368, −2.95393654252162424218046187240, −2.80212292510036994662328863548, −2.77783392857995085703289123041, −2.64997537838759852476012929644, −1.91814148541747434332632901686, −1.78781076218911119458511871537, −1.47874488893559954389493611207, −1.35896686894928774431925936307, −0.56557605447525627040454183607, −0.18336990947785650624021473923,
0.18336990947785650624021473923, 0.56557605447525627040454183607, 1.35896686894928774431925936307, 1.47874488893559954389493611207, 1.78781076218911119458511871537, 1.91814148541747434332632901686, 2.64997537838759852476012929644, 2.77783392857995085703289123041, 2.80212292510036994662328863548, 2.95393654252162424218046187240, 3.32725408469917901749656852368, 3.59958093013743889585048963707, 4.00924234978420940382886947876, 4.22470355973473536538549117117, 4.46101054022409817401880442905, 4.61929811004212363961206708831, 4.74802571157142289038999913584, 4.83705498833167358416491705276, 5.37244848859719355475287577929, 5.45134145812226616470449838336, 5.64739558181282096177381646039, 5.70958883213699101794291930542, 6.25456939991234159985501383387, 6.43441876828900572970375733728, 6.45395562642449660141746778741
