| L(s) = 1 | − 2·3-s + 3·9-s − 6·13-s + 16·17-s − 12·23-s − 25-s − 4·27-s − 16·29-s + 12·39-s − 8·43-s − 2·49-s − 32·51-s − 20·53-s − 4·61-s + 24·69-s + 2·75-s + 16·79-s + 5·81-s + 32·87-s − 8·101-s + 8·103-s − 32·113-s − 18·117-s + 6·121-s + 127-s + 16·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 1.66·13-s + 3.88·17-s − 2.50·23-s − 1/5·25-s − 0.769·27-s − 2.97·29-s + 1.92·39-s − 1.21·43-s − 2/7·49-s − 4.48·51-s − 2.74·53-s − 0.512·61-s + 2.88·69-s + 0.230·75-s + 1.80·79-s + 5/9·81-s + 3.43·87-s − 0.796·101-s + 0.788·103-s − 3.01·113-s − 1.66·117-s + 6/11·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4252942892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4252942892\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 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
−9.776812660539917868666646857624, −9.380868006985183524254671620338, −9.164648098992043126258251806747, −7.967755501812485561228285496236, −7.924516304471918249031822880989, −7.60157999218128904797367370205, −7.57931648032709487678338253195, −6.67829526072150589216804530312, −6.32165586001691345007767819040, −5.90305717372453475249963621835, −5.43522108211496359901707719781, −5.18065655367073685656284445920, −5.00983756711507803787381124442, −3.94229901643299151035543840278, −3.88333529438784479060681370999, −3.29015594267176494387254798236, −2.62699611274736641989469289528, −1.61995905749254872580956168927, −1.58719029553208689077713907126, −0.27417621537079751493502458474,
0.27417621537079751493502458474, 1.58719029553208689077713907126, 1.61995905749254872580956168927, 2.62699611274736641989469289528, 3.29015594267176494387254798236, 3.88333529438784479060681370999, 3.94229901643299151035543840278, 5.00983756711507803787381124442, 5.18065655367073685656284445920, 5.43522108211496359901707719781, 5.90305717372453475249963621835, 6.32165586001691345007767819040, 6.67829526072150589216804530312, 7.57931648032709487678338253195, 7.60157999218128904797367370205, 7.924516304471918249031822880989, 7.967755501812485561228285496236, 9.164648098992043126258251806747, 9.380868006985183524254671620338, 9.776812660539917868666646857624
