L(s) = 1 | − 2-s − 2·4-s + 3·5-s + 2·7-s + 3·8-s − 3·10-s − 2·13-s − 2·14-s + 16-s + 4·17-s − 6·20-s + 2·23-s − 2·25-s + 2·26-s − 4·28-s − 16·31-s − 2·32-s − 4·34-s + 6·35-s − 4·37-s + 9·40-s + 41-s − 2·43-s − 2·46-s − 47-s + 3·49-s + 2·50-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 4-s + 1.34·5-s + 0.755·7-s + 1.06·8-s − 0.948·10-s − 0.554·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s − 1.34·20-s + 0.417·23-s − 2/5·25-s + 0.392·26-s − 0.755·28-s − 2.87·31-s − 0.353·32-s − 0.685·34-s + 1.01·35-s − 0.657·37-s + 1.42·40-s + 0.156·41-s − 0.304·43-s − 0.294·46-s − 0.145·47-s + 3/7·49-s + 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 16 T + 121 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 51 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 13 T + 147 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 15 T + 163 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 181 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 178 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 131 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 15 T + 203 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 15 T + 233 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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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
−7.74607574496326732163732655815, −7.69268633393631913841493159831, −6.95629815061482692299567982049, −6.80293717043010550500974724616, −6.31253448537302011263069804746, −5.79973212324884226975447910529, −5.44421402194162496616611601776, −5.40949830285311885125023703007, −4.87855170042502645689574655224, −4.77250187209593489514959980178, −3.96634457488034567424463367243, −3.87219579206636883680054317876, −3.40356513863813462369537628487, −2.77788825050618481757192340986, −2.30442816032702091542250214980, −1.88414502760823338474179411097, −1.33183085835872365449060862032, −1.26301442173657144658398263786, 0, 0,
1.26301442173657144658398263786, 1.33183085835872365449060862032, 1.88414502760823338474179411097, 2.30442816032702091542250214980, 2.77788825050618481757192340986, 3.40356513863813462369537628487, 3.87219579206636883680054317876, 3.96634457488034567424463367243, 4.77250187209593489514959980178, 4.87855170042502645689574655224, 5.40949830285311885125023703007, 5.44421402194162496616611601776, 5.79973212324884226975447910529, 6.31253448537302011263069804746, 6.80293717043010550500974724616, 6.95629815061482692299567982049, 7.69268633393631913841493159831, 7.74607574496326732163732655815