| L(s) = 1 | − 2-s + 4-s − 3·7-s − 8-s − 2·9-s − 6·11-s + 3·14-s + 16-s + 2·18-s + 6·22-s + 11·23-s − 25-s − 3·28-s − 12·29-s − 32-s − 2·36-s + 6·37-s − 5·43-s − 6·44-s − 11·46-s + 2·49-s + 50-s + 2·53-s + 3·56-s + 12·58-s + 6·63-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.13·7-s − 0.353·8-s − 2/3·9-s − 1.80·11-s + 0.801·14-s + 1/4·16-s + 0.471·18-s + 1.27·22-s + 2.29·23-s − 1/5·25-s − 0.566·28-s − 2.22·29-s − 0.176·32-s − 1/3·36-s + 0.986·37-s − 0.762·43-s − 0.904·44-s − 1.62·46-s + 2/7·49-s + 0.141·50-s + 0.274·53-s + 0.400·56-s + 1.57·58-s + 0.755·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134848 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134848 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5739999973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5739999973\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 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
−9.352733033794665660016712838446, −9.087108842302983317496199614773, −8.250881820240080158760197930404, −8.047938829412603843644060308738, −7.45305948994342727376231437664, −6.95763591162711435474175464527, −6.52620063380190043504915162322, −5.84646355379168759586941761463, −5.27037225623674009934929085244, −5.07415766938746979480108578342, −3.88626641146340330119093914729, −3.22164735083811487510598227408, −2.78011245833562481602786710060, −2.08503333551468068918151050682, −0.55335193629733962147375192390,
0.55335193629733962147375192390, 2.08503333551468068918151050682, 2.78011245833562481602786710060, 3.22164735083811487510598227408, 3.88626641146340330119093914729, 5.07415766938746979480108578342, 5.27037225623674009934929085244, 5.84646355379168759586941761463, 6.52620063380190043504915162322, 6.95763591162711435474175464527, 7.45305948994342727376231437664, 8.047938829412603843644060308738, 8.250881820240080158760197930404, 9.087108842302983317496199614773, 9.352733033794665660016712838446
