| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 3·9-s + 2·13-s + 5·16-s − 8·17-s − 6·18-s + 14·19-s − 25-s + 4·26-s + 6·32-s − 16·34-s − 9·36-s + 28·38-s − 16·43-s + 18·47-s − 2·49-s − 2·50-s + 6·52-s − 22·53-s − 10·59-s + 7·64-s − 20·67-s − 24·68-s − 12·72-s + 42·76-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 9-s + 0.554·13-s + 5/4·16-s − 1.94·17-s − 1.41·18-s + 3.21·19-s − 1/5·25-s + 0.784·26-s + 1.06·32-s − 2.74·34-s − 3/2·36-s + 4.54·38-s − 2.43·43-s + 2.62·47-s − 2/7·49-s − 0.282·50-s + 0.832·52-s − 3.02·53-s − 1.30·59-s + 7/8·64-s − 2.44·67-s − 2.91·68-s − 1.41·72-s + 4.81·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.711749948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.711749948\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 17 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 141 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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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
−13.30155021929119161366332252046, −12.51617759768662949677062718120, −11.96265363250534614982759832729, −11.67626958654991442426548564908, −11.25111737501018010894859849339, −10.84344936356113208565349317874, −10.23045900052884807800347481520, −9.333687644170166982705742887466, −9.149973819366366417744698990508, −8.290761348738417328071218177964, −7.68616206475835792833706874124, −7.18802883405412538593069410516, −6.46767238726875828182751799868, −6.01869006498360600313377989302, −5.37230424282602979695769710811, −4.89538188592078517366076686232, −4.19873050916616556771392415588, −3.14050325071886133800571108651, −3.04581874417342468555685221023, −1.70330108837015106579750316817,
1.70330108837015106579750316817, 3.04581874417342468555685221023, 3.14050325071886133800571108651, 4.19873050916616556771392415588, 4.89538188592078517366076686232, 5.37230424282602979695769710811, 6.01869006498360600313377989302, 6.46767238726875828182751799868, 7.18802883405412538593069410516, 7.68616206475835792833706874124, 8.290761348738417328071218177964, 9.149973819366366417744698990508, 9.333687644170166982705742887466, 10.23045900052884807800347481520, 10.84344936356113208565349317874, 11.25111737501018010894859849339, 11.67626958654991442426548564908, 11.96265363250534614982759832729, 12.51617759768662949677062718120, 13.30155021929119161366332252046
