| L(s) = 1 | + 9·4-s − 4·5-s + 9·9-s + 90·11-s + 64·16-s + 270·19-s − 36·20-s + 125·25-s + 736·29-s − 340·31-s + 81·36-s + 988·41-s + 810·44-s − 36·45-s + 286·49-s − 360·55-s − 8·59-s − 524·61-s + 999·64-s − 4.55e3·71-s + 2.43e3·76-s − 124·79-s − 256·80-s − 2.41e3·89-s − 1.08e3·95-s + 810·99-s + 1.12e3·100-s + ⋯ |
| L(s) = 1 | + 9/8·4-s − 0.357·5-s + 1/3·9-s + 2.46·11-s + 16-s + 3.26·19-s − 0.402·20-s + 25-s + 4.71·29-s − 1.96·31-s + 3/8·36-s + 3.76·41-s + 2.77·44-s − 0.119·45-s + 0.833·49-s − 0.882·55-s − 0.0176·59-s − 1.09·61-s + 1.95·64-s − 7.60·71-s + 3.66·76-s − 0.176·79-s − 0.357·80-s − 2.87·89-s − 1.16·95-s + 0.822·99-s + 9/8·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.505210113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.505210113\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T - 109 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 286 T^{2} + p^{6} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - 9 T^{2} + 17 T^{4} - 9 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 45 T + 694 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 3025 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 6462 T^{2} + 17619875 T^{4} + 6462 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 135 T + 11366 T^{2} - 135 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 20365 T^{2} + 266697336 T^{4} + 20365 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 184 T + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 170 T - 891 T^{2} + 170 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 96265 T^{2} + 6701223816 T^{4} + 96265 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 247 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 100450 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 113843 T^{2} + 2181013320 T^{4} - 113843 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 295729 T^{2} + 65291280312 T^{4} + 295729 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 205363 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 262 T - 158337 T^{2} + 262 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 162350 T^{2} - 64100859669 T^{4} - 162350 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 1138 T + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 766370 T^{2} + 435988750611 T^{4} + 766370 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 62 T - 489195 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 937458 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 1206 T + 749467 T^{2} + 1206 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 24254 T^{2} + p^{6} T^{4} )^{2} \) |
| 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
−9.573325932325800565805575918598, −9.436421170167226449409314878336, −8.955197920804585691852598522289, −8.936878366601372895663272676518, −8.657584372759447094067262289412, −8.035896189564581621594601153064, −7.896503545722239773498890309907, −7.30090253369165016912246666507, −7.19912836814955642284616698659, −7.17313733947932755363815623258, −6.66974195148953556916776951463, −6.45075123315215320871426222173, −5.86183798865195934209940777078, −5.73711241418096496789958973955, −5.57495047385299272744789853284, −4.65843082667287493475857326712, −4.41481219152517174749296249440, −4.35099725222925725326566953064, −3.66469536531814784122355287298, −3.07214498318776039233368851756, −2.97703928999345220578159389220, −2.59974059057939210308644225453, −1.43721061311397685907556067212, −1.13750445235385940111755474844, −1.02707373728464075859609208493,
1.02707373728464075859609208493, 1.13750445235385940111755474844, 1.43721061311397685907556067212, 2.59974059057939210308644225453, 2.97703928999345220578159389220, 3.07214498318776039233368851756, 3.66469536531814784122355287298, 4.35099725222925725326566953064, 4.41481219152517174749296249440, 4.65843082667287493475857326712, 5.57495047385299272744789853284, 5.73711241418096496789958973955, 5.86183798865195934209940777078, 6.45075123315215320871426222173, 6.66974195148953556916776951463, 7.17313733947932755363815623258, 7.19912836814955642284616698659, 7.30090253369165016912246666507, 7.896503545722239773498890309907, 8.035896189564581621594601153064, 8.657584372759447094067262289412, 8.936878366601372895663272676518, 8.955197920804585691852598522289, 9.436421170167226449409314878336, 9.573325932325800565805575918598
