| L(s) = 1 | − 4·2-s − 3-s + 12·4-s − 9·5-s + 4·6-s − 2·7-s − 32·8-s + 9·9-s + 36·10-s − 17·11-s − 12·12-s + 8·14-s + 9·15-s + 80·16-s + 25·17-s − 36·18-s + 7·19-s − 108·20-s + 2·21-s + 68·22-s + 9·23-s + 32·24-s + 29·25-s − 26·27-s − 24·28-s − 36·30-s + 57·31-s + ⋯ |
| L(s) = 1 | − 2·2-s − 1/3·3-s + 3·4-s − 9/5·5-s + 2/3·6-s − 2/7·7-s − 4·8-s + 9-s + 18/5·10-s − 1.54·11-s − 12-s + 4/7·14-s + 3/5·15-s + 5·16-s + 1.47·17-s − 2·18-s + 7/19·19-s − 5.39·20-s + 2/21·21-s + 3.09·22-s + 9/23·23-s + 4/3·24-s + 1.15·25-s − 0.962·27-s − 6/7·28-s − 6/5·30-s + 1.83·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2957037822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2957037822\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | $C_2$ | \( 1 + 2 T + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 8 T^{2} + p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T + 168 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T + 336 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 7 T - 312 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 556 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1490 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 57 T + 2044 T^{2} - 57 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 15 T + 1444 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 87 T + 4732 T^{2} + 87 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 - p T + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 55 T - 456 T^{2} - 55 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 39 T + 4228 T^{2} - 39 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 17 T - 4200 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 119 T + 8832 T^{2} + 119 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 129 T + 11788 T^{2} - 129 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 71 T - 2880 T^{2} + 71 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
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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
−15.64162963855443003717631900362, −15.03693907958629444804486106852, −14.84216110503369131336556574266, −13.39609390185370597348341429945, −12.68728471432642379838181288645, −12.11485452206431359743471847798, −11.73449471971366651011079544095, −11.18828228464196321820902704917, −10.54188698329264581462303970459, −9.978493070682500060836092791285, −9.669941328944072545979456327052, −8.533955750148234926553709147054, −8.005478117488796877493615808269, −7.57869474807904611022942158324, −7.25559595834741959298588121959, −6.27667829687070658451074276547, −5.26638285427959323095025978372, −3.78923785712948819789823619625, −2.76162049414574367655546538429, −0.75302290757209116481621355117,
0.75302290757209116481621355117, 2.76162049414574367655546538429, 3.78923785712948819789823619625, 5.26638285427959323095025978372, 6.27667829687070658451074276547, 7.25559595834741959298588121959, 7.57869474807904611022942158324, 8.005478117488796877493615808269, 8.533955750148234926553709147054, 9.669941328944072545979456327052, 9.978493070682500060836092791285, 10.54188698329264581462303970459, 11.18828228464196321820902704917, 11.73449471971366651011079544095, 12.11485452206431359743471847798, 12.68728471432642379838181288645, 13.39609390185370597348341429945, 14.84216110503369131336556574266, 15.03693907958629444804486106852, 15.64162963855443003717631900362
