L(s) = 1 | − 2·5-s + 4·7-s − 2·11-s − 8·19-s + 8·23-s + 3·25-s + 4·29-s − 4·31-s − 8·35-s − 20·37-s + 4·41-s − 12·43-s + 8·47-s + 4·49-s − 4·53-s + 4·55-s + 4·59-s − 4·61-s − 8·67-s + 12·71-s − 16·73-s − 8·77-s − 16·79-s + 20·83-s − 12·89-s + 16·95-s + 4·97-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 1.51·7-s − 0.603·11-s − 1.83·19-s + 1.66·23-s + 3/5·25-s + 0.742·29-s − 0.718·31-s − 1.35·35-s − 3.28·37-s + 0.624·41-s − 1.82·43-s + 1.16·47-s + 4/7·49-s − 0.549·53-s + 0.539·55-s + 0.520·59-s − 0.512·61-s − 0.977·67-s + 1.42·71-s − 1.87·73-s − 0.911·77-s − 1.80·79-s + 2.19·83-s − 1.27·89-s + 1.64·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 7 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 116 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 98 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 204 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.50196893551743508311641120842, −7.39552976143730798028710004868, −7.13323736408460665996055281965, −6.72621337704793379006426835851, −6.36338135333306369921578497480, −5.96235729028925883425381193873, −5.29621207190286166254413426758, −5.17668656281360037494360256519, −4.88371939327483745401661212689, −4.60300688022148887735136521727, −4.01254405545711370278356962426, −3.92122848031816831228968099275, −3.20272342454359496617743819658, −3.04524449676728095869573041689, −2.36652994499655995678795707861, −2.02717044804712489857557900404, −1.43487672660619070492223704370, −1.17963226752299774990382012123, 0, 0,
1.17963226752299774990382012123, 1.43487672660619070492223704370, 2.02717044804712489857557900404, 2.36652994499655995678795707861, 3.04524449676728095869573041689, 3.20272342454359496617743819658, 3.92122848031816831228968099275, 4.01254405545711370278356962426, 4.60300688022148887735136521727, 4.88371939327483745401661212689, 5.17668656281360037494360256519, 5.29621207190286166254413426758, 5.96235729028925883425381193873, 6.36338135333306369921578497480, 6.72621337704793379006426835851, 7.13323736408460665996055281965, 7.39552976143730798028710004868, 7.50196893551743508311641120842