| L(s) = 1 | − 48·19-s − 152·31-s − 104·49-s − 280·61-s − 120·79-s − 296·109-s + 288·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 152·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 2.52·19-s − 4.90·31-s − 2.12·49-s − 4.59·61-s − 1.51·79-s − 2.71·109-s + 2.38·121-s + 0.00787·127-s + 0.00763·131-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 + 0.899·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8015814706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8015814706\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 144 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 76 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 210 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 42 p T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1610 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 38 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 2692 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 1440 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 902 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 1470 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 2798 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 4030 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 10474 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 4254 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 14784 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 9802 T^{2} + p^{4} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.97492945648656685065131804051, −6.79002217020855822221016092016, −6.71993439428011531682872726217, −6.23726847814968309774348929963, −6.03946445438410558535678368430, −6.03125911108660414006290905697, −5.77543101874315172434024204729, −5.43125646093775155182878869347, −5.14123772811048715190374210306, −4.88845811926074045578706830482, −4.79152775264693691450513299918, −4.38629594644032385999612074794, −4.15590735130681612774565201522, −3.83937486525708878971231047322, −3.80541482826874959778819830934, −3.41988504735916608520818354052, −3.01754997472860181795635429116, −2.88007375832609284570793818231, −2.54541287417446720637653757976, −1.93208644670924510571748527132, −1.84104342353497849701447755421, −1.61474841748215178196031196226, −1.42029980461925101796746548800, −0.32579053159555237919962059638, −0.27702016865333959954879709362,
0.27702016865333959954879709362, 0.32579053159555237919962059638, 1.42029980461925101796746548800, 1.61474841748215178196031196226, 1.84104342353497849701447755421, 1.93208644670924510571748527132, 2.54541287417446720637653757976, 2.88007375832609284570793818231, 3.01754997472860181795635429116, 3.41988504735916608520818354052, 3.80541482826874959778819830934, 3.83937486525708878971231047322, 4.15590735130681612774565201522, 4.38629594644032385999612074794, 4.79152775264693691450513299918, 4.88845811926074045578706830482, 5.14123772811048715190374210306, 5.43125646093775155182878869347, 5.77543101874315172434024204729, 6.03125911108660414006290905697, 6.03946445438410558535678368430, 6.23726847814968309774348929963, 6.71993439428011531682872726217, 6.79002217020855822221016092016, 6.97492945648656685065131804051
