| L(s) = 1 | + 2-s + 2·7-s + 8-s − 6·13-s + 2·14-s − 16-s + 9·17-s + 6·19-s + 9·23-s + 3·25-s − 6·26-s + 13·29-s + 2·31-s − 6·32-s + 9·34-s + 4·37-s + 6·38-s + 14·41-s − 7·43-s + 9·46-s − 7·47-s + 3·49-s + 3·50-s + 15·53-s + 2·56-s + 13·58-s − 17·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.755·7-s + 0.353·8-s − 1.66·13-s + 0.534·14-s − 1/4·16-s + 2.18·17-s + 1.37·19-s + 1.87·23-s + 3/5·25-s − 1.17·26-s + 2.41·29-s + 0.359·31-s − 1.06·32-s + 1.54·34-s + 0.657·37-s + 0.973·38-s + 2.18·41-s − 1.06·43-s + 1.32·46-s − 1.02·47-s + 3/7·49-s + 0.424·50-s + 2.06·53-s + 0.267·56-s + 1.70·58-s − 2.21·59-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
| \(L(1)\) |
\(\approx\) |
\(7.571236589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.571236589\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 3 p T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 63 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 13 T + 97 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 95 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 7 T + 77 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 15 T + 133 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 17 T + 187 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 125 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 53 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 145 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 13 T + 197 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 3 T - 83 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 181 T^{2} + p^{2} T^{4} \) |
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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
−7.961083481976143769329879098664, −7.50764694769195348260888316094, −7.39038235903413019365567364335, −7.23779013852072032589718516353, −6.48919547613271431608966067873, −6.47726932272172158291263531099, −5.83262485204259101669740118669, −5.31405551713313597778123941996, −5.07452459652330767978626851481, −5.02573954177818560766356826220, −4.61983898418802152104477359507, −4.34682444718137929669276101081, −3.68760918993441633333963852015, −3.28368868629168618940077015733, −3.00629541431640365540182576106, −2.50116160886032842779820104546, −2.29010494093887903760393971111, −1.34774191974826322371713560314, −1.03729060068105822185530304431, −0.71421799627400156027274158670,
0.71421799627400156027274158670, 1.03729060068105822185530304431, 1.34774191974826322371713560314, 2.29010494093887903760393971111, 2.50116160886032842779820104546, 3.00629541431640365540182576106, 3.28368868629168618940077015733, 3.68760918993441633333963852015, 4.34682444718137929669276101081, 4.61983898418802152104477359507, 5.02573954177818560766356826220, 5.07452459652330767978626851481, 5.31405551713313597778123941996, 5.83262485204259101669740118669, 6.47726932272172158291263531099, 6.48919547613271431608966067873, 7.23779013852072032589718516353, 7.39038235903413019365567364335, 7.50764694769195348260888316094, 7.961083481976143769329879098664
