| L(s) = 1 | + 4·5-s + 6·9-s − 8·23-s − 2·25-s + 8·31-s + 4·37-s − 2·41-s − 16·43-s + 24·45-s + 14·49-s − 36·61-s − 36·73-s + 25·81-s + 8·83-s − 32·103-s + 40·107-s − 20·113-s − 32·115-s + 46·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 32·155-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2·9-s − 1.66·23-s − 2/5·25-s + 1.43·31-s + 0.657·37-s − 0.312·41-s − 2.43·43-s + 3.57·45-s + 2·49-s − 4.60·61-s − 4.21·73-s + 25/9·81-s + 0.878·83-s − 3.15·103-s + 3.86·107-s − 1.88·113-s − 2.98·115-s + 4.18·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 2.57·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.334993158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.334993158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + 2 T - 25 T^{2} - 68 T^{3} - 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 3 | \( 1 - 2 p T^{2} + 11 T^{4} - 4 T^{6} + 11 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 5 | \( ( 1 - 2 T + 7 T^{2} - 12 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 7 | \( 1 - 2 p T^{2} + 3 p^{2} T^{4} - 20 p^{2} T^{6} + 3 p^{4} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 46 T^{2} + 1003 T^{4} - 13588 T^{6} + 1003 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 30 T^{2} + 359 T^{4} - 3076 T^{6} + 359 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 6 T + p T^{2} )^{3}( 1 + 6 T + p T^{2} )^{3} \) |
| 19 | \( 1 - 102 T^{2} + 4523 T^{4} - 111940 T^{6} + 4523 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 4 T + 37 T^{2} + 120 T^{3} + 37 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( 1 - 94 T^{2} + 4423 T^{4} - 146692 T^{6} + 4423 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 4 T + 61 T^{2} - 184 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 2 T + 103 T^{2} - 140 T^{3} + 103 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 8 T + 113 T^{2} + 624 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 102 T^{2} + 8723 T^{4} - 447940 T^{6} + 8723 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 110 T^{2} - 1129 T^{4} + 440412 T^{6} - 1129 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 65 T^{2} + 448 T^{3} + 65 p T^{4} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 18 T + 179 T^{2} + 1292 T^{3} + 179 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 270 T^{2} + 34043 T^{4} - 2737876 T^{6} + 34043 p^{2} T^{8} - 270 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 182 T^{2} + 21059 T^{4} - 1609188 T^{6} + 21059 p^{2} T^{8} - 182 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( ( 1 + 18 T + 299 T^{2} + 2732 T^{3} + 299 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 - 294 T^{2} + 46163 T^{4} - 4481092 T^{6} + 46163 p^{2} T^{8} - 294 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( ( 1 - 4 T + 105 T^{2} - 600 T^{3} + 105 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 + 10 T^{2} + 21407 T^{4} + 158988 T^{6} + 21407 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 134 T^{2} + 17487 T^{4} - 1090580 T^{6} + 17487 p^{2} T^{8} - 134 p^{4} T^{10} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.65386990832185531843022394147, −4.38455863946506737826793796275, −4.36078912055168659957200338213, −4.16373815029545199167018450778, −4.07466485012915508471956748550, −3.81713303808241432692117565178, −3.76548971057099029718569806705, −3.68779260048829180406081819448, −3.32697447937824561317429677899, −3.24613607304898982671415433464, −3.03417649180535340891097860618, −2.83270523918440489559983996371, −2.71070692018500024018848650779, −2.51391925123712000074502697954, −2.37950473844909759733374901229, −2.10251855941264294248091821629, −2.06963272841553113755734096282, −1.78643037874164764302713828425, −1.52860279079320962806512364447, −1.51959281417665503866402668859, −1.37685155812252076295832513297, −1.25584306845889122963700601550, −0.932321721038537633443064559327, −0.40500999712133078905093483600, −0.15899309110387317159667477260,
0.15899309110387317159667477260, 0.40500999712133078905093483600, 0.932321721038537633443064559327, 1.25584306845889122963700601550, 1.37685155812252076295832513297, 1.51959281417665503866402668859, 1.52860279079320962806512364447, 1.78643037874164764302713828425, 2.06963272841553113755734096282, 2.10251855941264294248091821629, 2.37950473844909759733374901229, 2.51391925123712000074502697954, 2.71070692018500024018848650779, 2.83270523918440489559983996371, 3.03417649180535340891097860618, 3.24613607304898982671415433464, 3.32697447937824561317429677899, 3.68779260048829180406081819448, 3.76548971057099029718569806705, 3.81713303808241432692117565178, 4.07466485012915508471956748550, 4.16373815029545199167018450778, 4.36078912055168659957200338213, 4.38455863946506737826793796275, 4.65386990832185531843022394147
Plot not available for L-functions of degree greater than 10.
