| L(s) = 1 | − 2·2-s − 3·3-s + 5·5-s + 6·6-s − 7·7-s + 8·8-s − 10·10-s + 6·11-s − 38·13-s + 14·14-s − 15·15-s − 16·16-s − 12·17-s − 119·19-s + 21·21-s − 12·22-s + 12·23-s − 24·24-s + 76·26-s + 27·27-s − 504·29-s + 30·30-s − 251·31-s − 18·33-s + 24·34-s − 35·35-s − 359·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.164·11-s − 0.810·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.171·17-s − 1.43·19-s + 0.218·21-s − 0.116·22-s + 0.108·23-s − 0.204·24-s + 0.573·26-s + 0.192·27-s − 3.22·29-s + 0.182·30-s − 1.45·31-s − 0.0949·33-s + 0.121·34-s − 0.169·35-s − 1.59·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.07846336960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07846336960\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 + p T + p^{3} T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 6 T - 1295 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 19 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 12 T - 4769 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 119 T + 7302 T^{2} + 119 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 12 T - 12023 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 252 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 251 T + 33210 T^{2} + 251 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 359 T + 78228 T^{2} + 359 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 37 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 246 T - 43307 T^{2} - 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 552 T + 155827 T^{2} + 552 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 408 T - 38915 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 386 T - 77985 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 811 T + 356958 T^{2} - 811 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 173 T - 359088 T^{2} + 173 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 1061 T + 632682 T^{2} + 1061 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 1206 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 672 T - 253385 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 818 T + p^{3} T^{2} )^{2} \) |
| 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
−12.24605804806141615925464283492, −11.41448343292958628405588627752, −11.16843100863399845208298673119, −10.60341044139215073622712442385, −10.32781214861380969413873303112, −9.510166308799271983269916679919, −9.306269961957255542749222419193, −8.932880389881737377583337981769, −8.231326747971920768511146371333, −7.38262141188688069155243485947, −7.36512121437007473707643839439, −6.43366501214109102995355399371, −6.06231974164739265204685998735, −5.33458899397421617367425424622, −4.90077048018799982323243371475, −3.98562462749233852090560606605, −3.39487003387775237210176178018, −2.14555124101629825149027709775, −1.70597629687504056218990116326, −0.13323528982947515330991007550,
0.13323528982947515330991007550, 1.70597629687504056218990116326, 2.14555124101629825149027709775, 3.39487003387775237210176178018, 3.98562462749233852090560606605, 4.90077048018799982323243371475, 5.33458899397421617367425424622, 6.06231974164739265204685998735, 6.43366501214109102995355399371, 7.36512121437007473707643839439, 7.38262141188688069155243485947, 8.231326747971920768511146371333, 8.932880389881737377583337981769, 9.306269961957255542749222419193, 9.510166308799271983269916679919, 10.32781214861380969413873303112, 10.60341044139215073622712442385, 11.16843100863399845208298673119, 11.41448343292958628405588627752, 12.24605804806141615925464283492
