| L(s) = 1 | + 4·3-s + 10·9-s + 6·13-s + 10·17-s − 2·23-s − 2·25-s + 20·27-s − 8·29-s + 24·39-s + 20·43-s + 15·49-s + 40·51-s − 2·53-s − 2·61-s − 8·69-s − 8·75-s − 10·79-s + 35·81-s − 32·87-s + 8·101-s + 24·103-s + 10·107-s − 16·113-s + 60·117-s + 35·121-s + 127-s + 80·129-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 10/3·9-s + 1.66·13-s + 2.42·17-s − 0.417·23-s − 2/5·25-s + 3.84·27-s − 1.48·29-s + 3.84·39-s + 3.04·43-s + 15/7·49-s + 5.60·51-s − 0.274·53-s − 0.256·61-s − 0.963·69-s − 0.923·75-s − 1.12·79-s + 35/9·81-s − 3.43·87-s + 0.796·101-s + 2.36·103-s + 0.966·107-s − 1.50·113-s + 5.54·117-s + 3.18·121-s + 0.0887·127-s + 7.04·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.98958283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.98958283\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 15 T^{2} + 116 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 35 T^{2} + 544 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 1054 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + T + 42 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 115 T^{2} + 5836 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 87 T^{2} + 5216 T^{4} - 87 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 + T + 68 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 7990 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 56 T^{2} + 4254 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 163 T^{2} + 15768 T^{4} - 163 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 208 T^{2} + 19774 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 5 T + 126 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 296 T^{2} + 35614 T^{4} - 296 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 335 T^{2} + 43792 T^{4} - 335 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 379 T^{2} + 54724 T^{4} - 379 p^{2} T^{6} + p^{4} T^{8} \) |
| 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
−7.12151633480150168634057872694, −6.34472503543136095429871376847, −6.17514903232155148876978524905, −6.09563594177782928929689385451, −6.03098227173383583197813891799, −5.62568690436271476179538796313, −5.42388169299074363350989532818, −5.40906282116556334276272930331, −4.78368970292020282905180182126, −4.70572209915381456133256806865, −4.24191734411024862170000301152, −4.16286404355042877168338383969, −4.06618285793661004751734627016, −3.54739189761823707952537163069, −3.43460018745110648319519130188, −3.39819438303702252786460812972, −3.25388519388819585016307901428, −2.73534645922557208625237661024, −2.44039755837552907615156142815, −2.30508140081603299010927607797, −1.94587392207633936538331016138, −1.68717261065427266987828045144, −1.20585832586164213704937753322, −1.03117964914297343101637268739, −0.65794469129480809042476282555,
0.65794469129480809042476282555, 1.03117964914297343101637268739, 1.20585832586164213704937753322, 1.68717261065427266987828045144, 1.94587392207633936538331016138, 2.30508140081603299010927607797, 2.44039755837552907615156142815, 2.73534645922557208625237661024, 3.25388519388819585016307901428, 3.39819438303702252786460812972, 3.43460018745110648319519130188, 3.54739189761823707952537163069, 4.06618285793661004751734627016, 4.16286404355042877168338383969, 4.24191734411024862170000301152, 4.70572209915381456133256806865, 4.78368970292020282905180182126, 5.40906282116556334276272930331, 5.42388169299074363350989532818, 5.62568690436271476179538796313, 6.03098227173383583197813891799, 6.09563594177782928929689385451, 6.17514903232155148876978524905, 6.34472503543136095429871376847, 7.12151633480150168634057872694
