| L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 2·6-s + 2·7-s − 4·8-s − 4·9-s − 5·11-s + 3·12-s − 13-s − 4·14-s + 5·16-s + 8·18-s + 2·21-s + 10·22-s + 23-s − 4·24-s + 2·26-s − 6·27-s + 6·28-s − 3·29-s + 17·31-s − 6·32-s − 5·33-s − 12·36-s + 2·37-s − 39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.816·6-s + 0.755·7-s − 1.41·8-s − 4/3·9-s − 1.50·11-s + 0.866·12-s − 0.277·13-s − 1.06·14-s + 5/4·16-s + 1.88·18-s + 0.436·21-s + 2.13·22-s + 0.208·23-s − 0.816·24-s + 0.392·26-s − 1.15·27-s + 1.13·28-s − 0.557·29-s + 3.05·31-s − 1.06·32-s − 0.870·33-s − 2·36-s + 0.328·37-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.067839036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067839036\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T + 35 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 3 T - T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 17 T + 133 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 17 T + 153 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 19 T + 211 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 123 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 117 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 59 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.207007149496274768084771497279, −9.163989806725291634246632017566, −8.460047583285389607433357308928, −8.345416302891367089728665455621, −7.85083522820591253615344971463, −7.83845619818879868059911359685, −7.43754249445562827816109448725, −6.77693211367059718523087150475, −6.30342451127997648623734366645, −5.98503522449555619853665348234, −5.36115706059252588801950825631, −5.23738465172214104059602453053, −4.47617856205482839765141515619, −4.07417128609313017282858734365, −3.12247841847445430566933046554, −2.84926902413657059545312335432, −2.42693996546770832138538433496, −2.16233851146825294382014338910, −1.11681020921238536563500437236, −0.51467502370781099341159803573,
0.51467502370781099341159803573, 1.11681020921238536563500437236, 2.16233851146825294382014338910, 2.42693996546770832138538433496, 2.84926902413657059545312335432, 3.12247841847445430566933046554, 4.07417128609313017282858734365, 4.47617856205482839765141515619, 5.23738465172214104059602453053, 5.36115706059252588801950825631, 5.98503522449555619853665348234, 6.30342451127997648623734366645, 6.77693211367059718523087150475, 7.43754249445562827816109448725, 7.83845619818879868059911359685, 7.85083522820591253615344971463, 8.345416302891367089728665455621, 8.460047583285389607433357308928, 9.163989806725291634246632017566, 9.207007149496274768084771497279
