| L(s) = 1 | + 16·7-s − 16·13-s + 64·19-s + 48·25-s + 80·31-s − 52·37-s + 32·43-s + 94·49-s − 108·61-s − 160·67-s + 192·73-s − 208·79-s − 256·91-s − 160·97-s − 144·103-s − 176·109-s + 114·121-s + 127-s + 131-s + 1.02e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 16/7·7-s − 1.23·13-s + 3.36·19-s + 1.91·25-s + 2.58·31-s − 1.40·37-s + 0.744·43-s + 1.91·49-s − 1.77·61-s − 2.38·67-s + 2.63·73-s − 2.63·79-s − 2.81·91-s − 1.64·97-s − 1.39·103-s − 1.61·109-s + 0.942·121-s + 0.00787·127-s + 0.00763·131-s + 7.69·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.248057172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.248057172\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 48 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 114 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 416 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 94 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 240 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 1056 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4290 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4560 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6450 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 54 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 3810 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 104 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3410 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 9792 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{2} \) |
| 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
−11.77573640306890491862472138972, −11.52911778306570364466573447617, −10.78239442851622589664447545251, −10.66683558921038300058287256799, −9.809451934059631004786224760246, −9.613536871876935525288933808240, −8.902707788714713351844101681885, −8.396358905066786162626619244616, −7.925303180359409807771799719426, −7.50691016726881283971362010970, −7.17710914941775217992666762092, −6.50633289204799949658122530871, −5.37943848611211427157673225897, −5.31927160319917222933418062191, −4.70485010934867225212261895938, −4.39837943542885904225736414052, −3.05674292188426112124169545645, −2.79127935295078644708489562469, −1.52132533619619718139131827775, −1.06233566860172633270219552727,
1.06233566860172633270219552727, 1.52132533619619718139131827775, 2.79127935295078644708489562469, 3.05674292188426112124169545645, 4.39837943542885904225736414052, 4.70485010934867225212261895938, 5.31927160319917222933418062191, 5.37943848611211427157673225897, 6.50633289204799949658122530871, 7.17710914941775217992666762092, 7.50691016726881283971362010970, 7.925303180359409807771799719426, 8.396358905066786162626619244616, 8.902707788714713351844101681885, 9.613536871876935525288933808240, 9.809451934059631004786224760246, 10.66683558921038300058287256799, 10.78239442851622589664447545251, 11.52911778306570364466573447617, 11.77573640306890491862472138972
