| L(s) = 1 | − 3·2-s + 4·4-s + 2·5-s − 5·7-s − 3·8-s − 3·9-s − 6·10-s − 12·13-s + 15·14-s + 3·16-s − 6·17-s + 9·18-s + 8·20-s + 3·25-s + 36·26-s − 20·28-s − 6·31-s − 6·32-s + 18·34-s − 10·35-s − 12·36-s + 2·37-s − 6·40-s − 6·41-s − 43-s − 6·45-s + 9·47-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s + 0.894·5-s − 1.88·7-s − 1.06·8-s − 9-s − 1.89·10-s − 3.32·13-s + 4.00·14-s + 3/4·16-s − 1.45·17-s + 2.12·18-s + 1.78·20-s + 3/5·25-s + 7.06·26-s − 3.77·28-s − 1.07·31-s − 1.06·32-s + 3.08·34-s − 1.69·35-s − 2·36-s + 0.328·37-s − 0.948·40-s − 0.937·41-s − 0.152·43-s − 0.894·45-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T + 65 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 21 T + 208 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T + 181 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 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
−11.04342682904046523127923700886, −10.67642057053853458444459246738, −10.05448170005347797617838680089, −10.04195209902878067160502177794, −9.257850046096451928541632099721, −9.204920602056123574059044303674, −9.136821973170020451015030571469, −8.277302939764079212594257057748, −7.59540456952920568550685745294, −7.28006356700913574440804444365, −6.74003389541514410650011968327, −6.28194559079649343755223877902, −5.61087585057795305472764425876, −5.08198454825342092572673501797, −4.24330873084334804408189828656, −2.90169475097090116190129486266, −2.77342669470078998389190329544, −1.93653161222263583938585826352, 0, 0,
1.93653161222263583938585826352, 2.77342669470078998389190329544, 2.90169475097090116190129486266, 4.24330873084334804408189828656, 5.08198454825342092572673501797, 5.61087585057795305472764425876, 6.28194559079649343755223877902, 6.74003389541514410650011968327, 7.28006356700913574440804444365, 7.59540456952920568550685745294, 8.277302939764079212594257057748, 9.136821973170020451015030571469, 9.204920602056123574059044303674, 9.257850046096451928541632099721, 10.04195209902878067160502177794, 10.05448170005347797617838680089, 10.67642057053853458444459246738, 11.04342682904046523127923700886
