| L(s) = 1 | − 2·3-s + 3·4-s − 4·5-s − 6·7-s + 2·9-s − 6·12-s + 8·15-s + 5·16-s + 8·17-s − 6·19-s − 12·20-s + 12·21-s + 11·25-s − 6·27-s − 18·28-s + 14·29-s + 6·31-s + 24·35-s + 6·36-s + 12·37-s − 8·45-s + 10·47-s − 10·48-s + 18·49-s − 16·51-s − 6·53-s + 12·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3/2·4-s − 1.78·5-s − 2.26·7-s + 2/3·9-s − 1.73·12-s + 2.06·15-s + 5/4·16-s + 1.94·17-s − 1.37·19-s − 2.68·20-s + 2.61·21-s + 11/5·25-s − 1.15·27-s − 3.40·28-s + 2.59·29-s + 1.07·31-s + 4.05·35-s + 36-s + 1.97·37-s − 1.19·45-s + 1.45·47-s − 1.44·48-s + 18/7·49-s − 2.24·51-s − 0.824·53-s + 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6152458215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6152458215\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 - 12 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 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
−12.68622784951561515177893531225, −12.19097081145056866600852743597, −12.04471604417061927192935491645, −11.32049532116622859071951103366, −11.15856402620335126794905674713, −10.40888189373920789067404749174, −10.13784840830060668236522434062, −9.685110375655533509715451054863, −8.755790042292369064489530030519, −8.006684532421444278784206510283, −7.64221805809677080574720146986, −7.01375535963790216563019695725, −6.53218383237364485054161855706, −6.18992396753207268461596818254, −5.79875921088286720469674759935, −4.60608325564846972986619446651, −4.00693683997329616539485261832, −3.04597537633662380246768635930, −2.89122140575968583395944552281, −0.75401270737500311514487533493,
0.75401270737500311514487533493, 2.89122140575968583395944552281, 3.04597537633662380246768635930, 4.00693683997329616539485261832, 4.60608325564846972986619446651, 5.79875921088286720469674759935, 6.18992396753207268461596818254, 6.53218383237364485054161855706, 7.01375535963790216563019695725, 7.64221805809677080574720146986, 8.006684532421444278784206510283, 8.755790042292369064489530030519, 9.685110375655533509715451054863, 10.13784840830060668236522434062, 10.40888189373920789067404749174, 11.15856402620335126794905674713, 11.32049532116622859071951103366, 12.04471604417061927192935491645, 12.19097081145056866600852743597, 12.68622784951561515177893531225
