| L(s) = 1 | + 3-s − 4-s − 7-s − 12-s − 13-s − 2·19-s − 21-s − 27-s + 28-s − 2·31-s − 2·37-s − 39-s + 2·43-s + 49-s + 52-s − 2·57-s − 2·61-s + 64-s − 67-s + 2·73-s + 2·76-s + 79-s − 81-s + 84-s + 91-s − 2·93-s + 2·97-s + ⋯ |
| L(s) = 1 | + 3-s − 4-s − 7-s − 12-s − 13-s − 2·19-s − 21-s − 27-s + 28-s − 2·31-s − 2·37-s − 39-s + 2·43-s + 49-s + 52-s − 2·57-s − 2·61-s + 64-s − 67-s + 2·73-s + 2·76-s + 79-s − 81-s + 84-s + 91-s − 2·93-s + 2·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7700625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7700625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4618342091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4618342091\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−9.031843230757328416849455948173, −9.015183814172875906982203347407, −8.534510650259875331456856028936, −8.177945297523974927319318121068, −7.74627459868093024716454330089, −7.25752970631518652623942107614, −7.12553867980214936885844345084, −6.44907272336912546633475734151, −6.26866911812278662721474622289, −5.54766927616507399365238336020, −5.41660700800335805744919894404, −4.81237065402472240080702292012, −4.27160523143446206874262988361, −4.08143887819308575331383866170, −3.55545717034202375835510797121, −3.20062985665109973672672817515, −2.62291206743571895402890240925, −2.11788227632453027801655877694, −1.81242571806638585162383979328, −0.38376211804009151934508624842,
0.38376211804009151934508624842, 1.81242571806638585162383979328, 2.11788227632453027801655877694, 2.62291206743571895402890240925, 3.20062985665109973672672817515, 3.55545717034202375835510797121, 4.08143887819308575331383866170, 4.27160523143446206874262988361, 4.81237065402472240080702292012, 5.41660700800335805744919894404, 5.54766927616507399365238336020, 6.26866911812278662721474622289, 6.44907272336912546633475734151, 7.12553867980214936885844345084, 7.25752970631518652623942107614, 7.74627459868093024716454330089, 8.177945297523974927319318121068, 8.534510650259875331456856028936, 9.015183814172875906982203347407, 9.031843230757328416849455948173
