| L(s) = 1 | + 8·5-s + 10·9-s + 40·13-s − 20·17-s − 2·25-s + 40·29-s + 40·37-s + 60·41-s + 80·45-s − 30·49-s − 120·53-s − 56·61-s + 320·65-s + 20·73-s + 19·81-s − 160·85-s − 44·89-s + 300·97-s − 280·101-s + 136·109-s − 380·113-s + 400·117-s + 42·121-s − 344·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 10/9·9-s + 3.07·13-s − 1.17·17-s − 0.0799·25-s + 1.37·29-s + 1.08·37-s + 1.46·41-s + 16/9·45-s − 0.612·49-s − 2.26·53-s − 0.918·61-s + 4.92·65-s + 0.273·73-s + 0.234·81-s − 1.88·85-s − 0.494·89-s + 3.09·97-s − 2.77·101-s + 1.24·109-s − 3.36·113-s + 3.41·117-s + 0.347·121-s − 2.75·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.677716551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.677716551\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 10 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 30 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 42 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 522 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 930 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3690 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 60 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5162 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2250 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6882 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 318 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13130 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 150 T + p^{2} T^{2} )^{2} \) |
| 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
−12.04172954566479108800306023449, −11.33767747918520609825770886740, −10.98166443150459818461311811844, −10.67505969302077752421677249641, −10.08826254634425439946180062947, −9.657364564751444080348145563373, −9.079604766500004422987308144880, −8.936094559713635220906035137268, −8.002082654717641475838572434030, −7.88381472139093370686889229571, −6.62667773009423432727076158072, −6.54013016591252017294481574878, −6.02769479051976688583780327371, −5.67045760106730213726884621195, −4.64754952032185163486173554796, −4.21029422508208155626960758624, −3.50619247641936555954293407402, −2.54884860889117592270814041170, −1.64089063735311132185522393751, −1.19934709933548363333336668148,
1.19934709933548363333336668148, 1.64089063735311132185522393751, 2.54884860889117592270814041170, 3.50619247641936555954293407402, 4.21029422508208155626960758624, 4.64754952032185163486173554796, 5.67045760106730213726884621195, 6.02769479051976688583780327371, 6.54013016591252017294481574878, 6.62667773009423432727076158072, 7.88381472139093370686889229571, 8.002082654717641475838572434030, 8.936094559713635220906035137268, 9.079604766500004422987308144880, 9.657364564751444080348145563373, 10.08826254634425439946180062947, 10.67505969302077752421677249641, 10.98166443150459818461311811844, 11.33767747918520609825770886740, 12.04172954566479108800306023449
