L(s) = 1 | − 4·4-s + 2·7-s − 5·9-s − 6·11-s + 10·13-s + 12·16-s + 6·17-s + 25-s − 8·28-s + 6·29-s + 20·36-s + 4·37-s − 20·43-s + 24·44-s + 18·47-s + 3·49-s − 40·52-s + 12·53-s − 10·63-s − 32·64-s − 24·68-s − 12·77-s + 16·81-s − 24·89-s + 20·91-s − 2·97-s + 30·99-s + ⋯ |
L(s) = 1 | − 2·4-s + 0.755·7-s − 5/3·9-s − 1.80·11-s + 2.77·13-s + 3·16-s + 1.45·17-s + 1/5·25-s − 1.51·28-s + 1.11·29-s + 10/3·36-s + 0.657·37-s − 3.04·43-s + 3.61·44-s + 2.62·47-s + 3/7·49-s − 5.54·52-s + 1.64·53-s − 1.25·63-s − 4·64-s − 2.91·68-s − 1.36·77-s + 16/9·81-s − 2.54·89-s + 2.09·91-s − 0.203·97-s + 3.01·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3441025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3441025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 53 | $C_2$ | \( 1 - 12 T + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−7.61351728406422952471450498282, −6.88163134068905072786790130896, −6.15165288423916240340816365337, −5.76794291139868174124926619278, −5.47907564040983295563221596641, −5.42303052201623691503191718514, −4.75114534169058008060760688384, −4.40797342389194916116282882235, −3.61689003045514298236484607491, −3.55356439922122982607524011465, −2.98751871897824111763777498176, −2.42156521546257849762441146583, −1.22672678124078466587000691964, −0.956984510621499067485306120141, 0,
0.956984510621499067485306120141, 1.22672678124078466587000691964, 2.42156521546257849762441146583, 2.98751871897824111763777498176, 3.55356439922122982607524011465, 3.61689003045514298236484607491, 4.40797342389194916116282882235, 4.75114534169058008060760688384, 5.42303052201623691503191718514, 5.47907564040983295563221596641, 5.76794291139868174124926619278, 6.15165288423916240340816365337, 6.88163134068905072786790130896, 7.61351728406422952471450498282