| L(s) = 1 | + 2·2-s − 4-s − 8·8-s + 2·9-s − 2·13-s − 7·16-s + 2·17-s + 4·18-s + 16·19-s + 6·25-s − 4·26-s + 14·32-s + 4·34-s − 2·36-s + 32·38-s − 8·43-s − 8·47-s + 14·49-s + 12·50-s + 2·52-s + 4·53-s − 16·59-s + 35·64-s − 8·67-s − 2·68-s − 16·72-s − 16·76-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s + 2/3·9-s − 0.554·13-s − 7/4·16-s + 0.485·17-s + 0.942·18-s + 3.67·19-s + 6/5·25-s − 0.784·26-s + 2.47·32-s + 0.685·34-s − 1/3·36-s + 5.19·38-s − 1.21·43-s − 1.16·47-s + 2·49-s + 1.69·50-s + 0.277·52-s + 0.549·53-s − 2.08·59-s + 35/8·64-s − 0.977·67-s − 0.242·68-s − 1.88·72-s − 1.83·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48841 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48841 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.991540589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.991540589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−12.59566083378210742826865264822, −12.17515416663513620558996049734, −11.72120186931329960919213248393, −11.59058139052802506104336248401, −10.48620505760998326838569995995, −9.828747935372716029753200366877, −9.818846342213870054422795426746, −8.991221120621212505531843607925, −8.900403962196435046218316034675, −7.88375998289842250682232661970, −7.47911440404715108785774339056, −6.90444905786425497934493499627, −6.05153822256967727854534173395, −5.51571643314335833471285867487, −5.02601559772614000376654132131, −4.76761372781537448004791781836, −3.96147137343355908357418248454, −3.13422340304376000753554160721, −3.07064655888383687928589829176, −1.09259821033513473568650343883,
1.09259821033513473568650343883, 3.07064655888383687928589829176, 3.13422340304376000753554160721, 3.96147137343355908357418248454, 4.76761372781537448004791781836, 5.02601559772614000376654132131, 5.51571643314335833471285867487, 6.05153822256967727854534173395, 6.90444905786425497934493499627, 7.47911440404715108785774339056, 7.88375998289842250682232661970, 8.900403962196435046218316034675, 8.991221120621212505531843607925, 9.818846342213870054422795426746, 9.828747935372716029753200366877, 10.48620505760998326838569995995, 11.59058139052802506104336248401, 11.72120186931329960919213248393, 12.17515416663513620558996049734, 12.59566083378210742826865264822
