| L(s) = 1 | + 4-s − 8·7-s − 2·9-s + 12·11-s + 4·13-s + 16-s − 2·17-s − 10·25-s − 8·28-s − 2·36-s − 8·37-s + 16·43-s + 12·44-s + 34·49-s + 4·52-s − 6·53-s + 16·63-s + 64-s − 2·68-s − 96·77-s − 5·81-s − 12·89-s − 32·91-s + 28·97-s − 24·99-s − 10·100-s − 12·107-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.02·7-s − 2/3·9-s + 3.61·11-s + 1.10·13-s + 1/4·16-s − 0.485·17-s − 2·25-s − 1.51·28-s − 1/3·36-s − 1.31·37-s + 2.43·43-s + 1.80·44-s + 34/7·49-s + 0.554·52-s − 0.824·53-s + 2.01·63-s + 1/8·64-s − 0.242·68-s − 10.9·77-s − 5/9·81-s − 1.27·89-s − 3.35·91-s + 2.84·97-s − 2.41·99-s − 100-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3247204 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3247204 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| 53 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−7.13690330956485355862455539993, −6.65330731261113455462227374582, −6.51750679489306161330527852278, −6.07940975059845120120491473895, −5.99911855171913780844071238816, −5.64127450416579378652046822039, −4.49730750500575581109364219588, −3.90229547122613769308947697962, −3.80320802252311147126867436038, −3.49677870190616727260642617802, −3.01941537763985245794009266319, −2.33242998394544666678362686278, −1.60618933200325615024936212313, −0.961208165563165598163043574434, 0,
0.961208165563165598163043574434, 1.60618933200325615024936212313, 2.33242998394544666678362686278, 3.01941537763985245794009266319, 3.49677870190616727260642617802, 3.80320802252311147126867436038, 3.90229547122613769308947697962, 4.49730750500575581109364219588, 5.64127450416579378652046822039, 5.99911855171913780844071238816, 6.07940975059845120120491473895, 6.51750679489306161330527852278, 6.65330731261113455462227374582, 7.13690330956485355862455539993
