| L(s) = 1 | + 3·2-s + 4·4-s − 2·5-s − 2·7-s + 3·8-s − 6·10-s + 4·13-s − 6·14-s + 3·16-s + 4·17-s + 2·19-s − 8·20-s − 8·23-s − 7·25-s + 12·26-s − 8·28-s + 8·29-s + 8·31-s + 6·32-s + 12·34-s + 4·35-s − 2·37-s + 6·38-s − 6·40-s + 8·43-s − 24·46-s − 4·47-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2·4-s − 0.894·5-s − 0.755·7-s + 1.06·8-s − 1.89·10-s + 1.10·13-s − 1.60·14-s + 3/4·16-s + 0.970·17-s + 0.458·19-s − 1.78·20-s − 1.66·23-s − 7/5·25-s + 2.35·26-s − 1.51·28-s + 1.48·29-s + 1.43·31-s + 1.06·32-s + 2.05·34-s + 0.676·35-s − 0.328·37-s + 0.973·38-s − 0.948·40-s + 1.21·43-s − 3.53·46-s − 0.583·47-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
| \(L(1)\) |
\(\approx\) |
\(6.452570005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.452570005\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 69 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 93 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 20 T + 186 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 9 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 24 T + 285 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−7.986357233864017206192692994727, −7.70556721209690893106970000191, −7.35142805070095427810826433065, −6.78520936500969628881143923525, −6.33020505163478152044122650853, −6.18886365752852979597928248040, −5.86213183094736430735593879584, −5.78774729073140963319274121558, −4.99663478025028643380481163859, −4.84400658006040156991923130204, −4.41531190092191203063922812696, −4.19107084511516130055754411668, −3.70497609526592559529554046116, −3.48254962602654026283734873520, −3.16782857261458040449854942769, −2.95992652847094003767598216851, −2.18306623022011482760887167795, −1.74018656696029237121681411715, −1.04212598280575554553967687476, −0.46065198705447933483692630233,
0.46065198705447933483692630233, 1.04212598280575554553967687476, 1.74018656696029237121681411715, 2.18306623022011482760887167795, 2.95992652847094003767598216851, 3.16782857261458040449854942769, 3.48254962602654026283734873520, 3.70497609526592559529554046116, 4.19107084511516130055754411668, 4.41531190092191203063922812696, 4.84400658006040156991923130204, 4.99663478025028643380481163859, 5.78774729073140963319274121558, 5.86213183094736430735593879584, 6.18886365752852979597928248040, 6.33020505163478152044122650853, 6.78520936500969628881143923525, 7.35142805070095427810826433065, 7.70556721209690893106970000191, 7.986357233864017206192692994727
