| L(s) = 1 | + 8·9-s − 6·11-s + 6·19-s + 2·29-s − 6·31-s − 18·41-s + 32·49-s + 20·59-s − 42·61-s + 2·71-s + 40·79-s + 24·81-s − 8·89-s − 48·99-s − 32·101-s − 14·109-s − 35·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 69·169-s + ⋯ |
| L(s) = 1 | + 8/3·9-s − 1.80·11-s + 1.37·19-s + 0.371·29-s − 1.07·31-s − 2.81·41-s + 32/7·49-s + 2.60·59-s − 5.37·61-s + 0.237·71-s + 4.50·79-s + 8/3·81-s − 0.847·89-s − 4.82·99-s − 3.18·101-s − 1.34·109-s − 3.18·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.30·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.280111552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.280111552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 - T )^{6} \) |
| good | 3 | \( 1 - 8 T^{2} + 40 T^{4} - 47 p T^{6} + 40 p^{2} T^{8} - 8 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 32 T^{2} + 480 T^{4} - 4261 T^{6} + 480 p^{2} T^{8} - 32 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 3 T + 31 T^{2} + 65 T^{3} + 31 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 69 T^{2} + 2089 T^{4} - 35377 T^{6} + 2089 p^{2} T^{8} - 69 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 60 T^{2} + 2020 T^{4} - 41801 T^{6} + 2020 p^{2} T^{8} - 60 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 120 T^{2} + 12 p^{2} T^{4} - 189357 T^{6} + 12 p^{4} T^{8} - 120 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - T + 9 T^{2} - 107 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 3 T + 57 T^{2} + 105 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 173 T^{2} + 377 p T^{4} - 657833 T^{6} + 377 p^{3} T^{8} - 173 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 9 T + 103 T^{2} + 563 T^{3} + 103 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 175 T^{2} + 15366 T^{4} - 820571 T^{6} + 15366 p^{2} T^{8} - 175 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 173 T^{2} + 13777 T^{4} - 738289 T^{6} + 13777 p^{2} T^{8} - 173 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 251 T^{2} + 28142 T^{4} - 1870607 T^{6} + 28142 p^{2} T^{8} - 251 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 10 T + 193 T^{2} - 1172 T^{3} + 193 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 21 T + 265 T^{2} + 37 p T^{3} + 265 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 305 T^{2} + 41985 T^{4} - 3488601 T^{6} + 41985 p^{2} T^{8} - 305 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - T + 132 T^{2} + 139 T^{3} + 132 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 65 T^{2} + 9333 T^{4} - 292257 T^{6} + 9333 p^{2} T^{8} - 65 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 20 T + 292 T^{2} - 2859 T^{3} + 292 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 261 T^{2} + 38793 T^{4} - 3975273 T^{6} + 38793 p^{2} T^{8} - 261 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 4 T + 148 T^{2} + 1067 T^{3} + 148 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 429 T^{2} + 86457 T^{4} - 10512313 T^{6} + 86457 p^{2} T^{8} - 429 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.30421526036297072065023350168, −4.25921993980672892949589834630, −4.14169948379037910479312859877, −4.05181969328075046486303025632, −3.69211282472426697328912494887, −3.60662581388110246341105251411, −3.55297171408828834017597842906, −3.52026849561204917171305787522, −3.27285231820678467931274711395, −3.00900683175834972874577745819, −2.90061353124348997683568267497, −2.63909753318428756730844710575, −2.60205291545495832822974439348, −2.40862489649484407571771670361, −2.25296497467054431748179219903, −2.21941951284794359741144319074, −1.83616106002060038410262931476, −1.58310955005510152360629385083, −1.57440518974004092979092703762, −1.34930806915059113116715711415, −1.16512632568380846914502097557, −1.14088560416699761886137482040, −0.61399921636274559247474804549, −0.47451669956612729669134553388, −0.23519039623746656479886125353,
0.23519039623746656479886125353, 0.47451669956612729669134553388, 0.61399921636274559247474804549, 1.14088560416699761886137482040, 1.16512632568380846914502097557, 1.34930806915059113116715711415, 1.57440518974004092979092703762, 1.58310955005510152360629385083, 1.83616106002060038410262931476, 2.21941951284794359741144319074, 2.25296497467054431748179219903, 2.40862489649484407571771670361, 2.60205291545495832822974439348, 2.63909753318428756730844710575, 2.90061353124348997683568267497, 3.00900683175834972874577745819, 3.27285231820678467931274711395, 3.52026849561204917171305787522, 3.55297171408828834017597842906, 3.60662581388110246341105251411, 3.69211282472426697328912494887, 4.05181969328075046486303025632, 4.14169948379037910479312859877, 4.25921993980672892949589834630, 4.30421526036297072065023350168
Plot not available for L-functions of degree greater than 10.
