| L(s) = 1 | − 2·5-s + 2·9-s + 4·13-s − 4·17-s + 3·25-s + 12·29-s − 20·37-s − 4·41-s − 4·45-s + 12·53-s + 4·61-s − 8·65-s + 12·73-s − 5·81-s + 8·85-s − 20·89-s − 4·97-s + 4·101-s − 36·109-s + 4·113-s + 8·117-s + 10·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2/3·9-s + 1.10·13-s − 0.970·17-s + 3/5·25-s + 2.22·29-s − 3.28·37-s − 0.624·41-s − 0.596·45-s + 1.64·53-s + 0.512·61-s − 0.992·65-s + 1.40·73-s − 5/9·81-s + 0.867·85-s − 2.11·89-s − 0.406·97-s + 0.398·101-s − 3.44·109-s + 0.376·113-s + 0.739·117-s + 0.909·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61465600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61465600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.643071126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.643071126\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.096242190930961005146510536896, −7.77087821762918442200365577815, −7.14654683277111244040270936452, −6.89167183018779236315471616679, −6.76235354175399515146102245361, −6.57687824691543518859803743633, −5.94069436733095479721407798336, −5.51296754160000869575138200713, −5.24306653916822529077021888473, −4.68390429206025435969984102269, −4.59041459641937266322808744384, −3.99438949311580279276895636010, −3.65649941538504025037980472022, −3.61377616290315211883575436331, −2.75942028832061197332933407869, −2.69535297313801770547697474098, −1.89202836879679022695383497498, −1.49873079156037818008183947662, −1.01423341140987054578483275475, −0.34369546063345045483851833980,
0.34369546063345045483851833980, 1.01423341140987054578483275475, 1.49873079156037818008183947662, 1.89202836879679022695383497498, 2.69535297313801770547697474098, 2.75942028832061197332933407869, 3.61377616290315211883575436331, 3.65649941538504025037980472022, 3.99438949311580279276895636010, 4.59041459641937266322808744384, 4.68390429206025435969984102269, 5.24306653916822529077021888473, 5.51296754160000869575138200713, 5.94069436733095479721407798336, 6.57687824691543518859803743633, 6.76235354175399515146102245361, 6.89167183018779236315471616679, 7.14654683277111244040270936452, 7.77087821762918442200365577815, 8.096242190930961005146510536896
