| L(s) = 1 | + 2-s + 4-s + 8-s − 8·13-s + 16-s + 25-s − 8·26-s − 16·31-s + 5·32-s − 16·37-s + 4·41-s + 18·49-s + 50-s − 8·52-s − 24·53-s − 16·62-s + 64-s − 16·71-s − 16·74-s + 16·79-s + 4·82-s + 16·83-s + 20·89-s + 18·98-s + 100-s − 8·104-s − 24·106-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2.21·13-s + 1/4·16-s + 1/5·25-s − 1.56·26-s − 2.87·31-s + 0.883·32-s − 2.63·37-s + 0.624·41-s + 18/7·49-s + 0.141·50-s − 1.10·52-s − 3.29·53-s − 2.03·62-s + 1/8·64-s − 1.89·71-s − 1.85·74-s + 1.80·79-s + 0.441·82-s + 1.75·83-s + 2.11·89-s + 1.81·98-s + 1/10·100-s − 0.784·104-s − 2.33·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.132633625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.132633625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T - p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T^{2} + 8 T^{3} - p T^{4} + p^{3} T^{6} \) |
| good | 7 | \( 1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 191 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 503 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 4 T + 23 T^{2} + 48 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 2255 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 1367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 2 p T^{2} + 1775 T^{4} - 40932 T^{6} + 1775 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 3207 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 8 T + 89 T^{2} + 432 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + 8 T + 3 p T^{2} + 584 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 - 2 T + 23 T^{2} - 220 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 65 T^{2} + 64 T^{3} + 65 p T^{4} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 22367 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 12 T + 191 T^{2} + 1264 T^{3} + 191 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 20567 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 20039 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 137 T^{2} - 64 T^{3} + 137 p T^{4} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 + 8 T + 133 T^{2} + 1008 T^{3} + 133 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 2367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 8 T + 233 T^{2} - 1200 T^{3} + 233 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 - 8 T + 185 T^{2} - 880 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 - 10 T + 103 T^{2} - 396 T^{3} + 103 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 39183 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.43139867470728115367100569362, −5.86090456268283388081922523685, −5.76455629105744345379488937079, −5.59010442374956402349068876320, −5.44105061759257269436399407651, −5.41877506110159868588410946213, −5.21524936965601010373350502873, −4.82974114920931660372560045327, −4.65905303233235395056769399932, −4.54964378579797929068876697320, −4.40152021646059700186722445524, −4.36347047387001100555464289813, −3.92126312093008078809157971821, −3.43813584515417679861540403641, −3.43687726664174114817933444810, −3.43502355762640642153229140711, −3.19112753406882920780044722043, −2.78242270731745568453443177115, −2.59374407491759750383671509794, −2.19010116685950845437714571762, −1.96040682753129733989406636289, −1.85786356006178278370954896093, −1.70717648325546091317542950325, −0.889785387306774308798492493614, −0.39259526158820347161299115304,
0.39259526158820347161299115304, 0.889785387306774308798492493614, 1.70717648325546091317542950325, 1.85786356006178278370954896093, 1.96040682753129733989406636289, 2.19010116685950845437714571762, 2.59374407491759750383671509794, 2.78242270731745568453443177115, 3.19112753406882920780044722043, 3.43502355762640642153229140711, 3.43687726664174114817933444810, 3.43813584515417679861540403641, 3.92126312093008078809157971821, 4.36347047387001100555464289813, 4.40152021646059700186722445524, 4.54964378579797929068876697320, 4.65905303233235395056769399932, 4.82974114920931660372560045327, 5.21524936965601010373350502873, 5.41877506110159868588410946213, 5.44105061759257269436399407651, 5.59010442374956402349068876320, 5.76455629105744345379488937079, 5.86090456268283388081922523685, 6.43139867470728115367100569362
Plot not available for L-functions of degree greater than 10.
