| L(s) = 1 | − 2·2-s + 3·3-s + 14·5-s − 6·6-s − 16·7-s + 8·8-s − 28·10-s + 64·11-s + 91·13-s + 32·14-s + 42·15-s − 16·16-s + 9·17-s + 72·19-s − 48·21-s − 128·22-s + 92·23-s + 24·24-s − 103·25-s − 182·26-s − 27·27-s + 113·29-s − 84·30-s − 448·31-s + 192·33-s − 18·34-s − 224·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1.25·5-s − 0.408·6-s − 0.863·7-s + 0.353·8-s − 0.885·10-s + 1.75·11-s + 1.94·13-s + 0.610·14-s + 0.722·15-s − 1/4·16-s + 0.128·17-s + 0.869·19-s − 0.498·21-s − 1.24·22-s + 0.834·23-s + 0.204·24-s − 0.823·25-s − 1.37·26-s − 0.192·27-s + 0.723·29-s − 0.511·30-s − 2.59·31-s + 1.01·33-s − 0.0907·34-s − 1.08·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.062134846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.062134846\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 p T + p^{3} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 7 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 16 T - 87 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 64 T + 2765 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T - 4832 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 72 T - 1675 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 p T - 7 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 113 T - 11620 T^{2} - 113 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 279 T + 27188 T^{2} + 279 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 387 T + 80848 T^{2} + 387 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 260 T - 11907 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 112 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 471 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 380 T - 60979 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 317 T - 126492 T^{2} - 317 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 260 T - 233163 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 64 T - 353815 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 1141 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 884 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 1428 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 282 T - 625445 T^{2} + 282 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 478 T - 684189 T^{2} - 478 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−13.93839594929733673986969757221, −13.75377282371362113359437808762, −13.34274442320439422054275604436, −12.84236440306736135521092467982, −11.82549785865952239648210887911, −11.60497905318121882673113195926, −10.54053967650488738551940864355, −10.33397228712344648557891156016, −9.420147556650467775899477181934, −9.117017808621916737849452082015, −8.955761367513565561474573213698, −8.152705681751948714075780117307, −7.11469753889560962401553593987, −6.67372646513405622481783292753, −5.90834400491222093780113635394, −5.40853523711170565167411276840, −3.67893202344513705564231806333, −3.63579404644443052206828074974, −1.98952355094789828322971855425, −1.13657477173618620595454948549,
1.13657477173618620595454948549, 1.98952355094789828322971855425, 3.63579404644443052206828074974, 3.67893202344513705564231806333, 5.40853523711170565167411276840, 5.90834400491222093780113635394, 6.67372646513405622481783292753, 7.11469753889560962401553593987, 8.152705681751948714075780117307, 8.955761367513565561474573213698, 9.117017808621916737849452082015, 9.420147556650467775899477181934, 10.33397228712344648557891156016, 10.54053967650488738551940864355, 11.60497905318121882673113195926, 11.82549785865952239648210887911, 12.84236440306736135521092467982, 13.34274442320439422054275604436, 13.75377282371362113359437808762, 13.93839594929733673986969757221
