| L(s) = 1 | + 4·3-s + 8·9-s − 12·13-s − 16-s + 8·25-s + 12·27-s − 4·31-s − 20·37-s − 48·39-s + 32·43-s − 4·48-s − 32·61-s − 4·67-s + 8·73-s + 32·75-s + 23·81-s − 16·93-s + 12·97-s + 40·103-s − 80·111-s − 96·117-s + 8·121-s + 127-s + 128·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s − 3.32·13-s − 1/4·16-s + 8/5·25-s + 2.30·27-s − 0.718·31-s − 3.28·37-s − 7.68·39-s + 4.87·43-s − 0.577·48-s − 4.09·61-s − 0.488·67-s + 0.936·73-s + 3.69·75-s + 23/9·81-s − 1.65·93-s + 1.21·97-s + 3.94·103-s − 7.59·111-s − 8.87·117-s + 8/11·121-s + 0.0887·127-s + 11.2·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.539086211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.539086211\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 1054 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 718 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.41384087811808677296044834966, −7.11733374346453031684230475801, −7.08232546690759435107189515270, −6.65067766531095354132003118116, −6.24219707712050805445023914349, −6.20193679202515745442359201255, −6.02703724300821650857947394985, −5.43590190153375436318456036634, −5.15497109352716466242633866706, −5.01503037967307435874493270303, −4.98184243954086951435181798117, −4.56650918508809601958237593532, −4.44082368979591677309874429857, −3.94653290967398345098659463652, −3.89076450321693730876667080315, −3.50464013593630016629411799984, −3.10199337933578825288728933139, −3.02860342749846366064179571852, −2.62623386413265361043416781078, −2.51449263104060758167431677344, −2.34924541118969268871658147381, −1.78449513312604381026496152832, −1.77191542487634168713594702616, −1.03492709954164932533007146838, −0.35845225930331716462255307627,
0.35845225930331716462255307627, 1.03492709954164932533007146838, 1.77191542487634168713594702616, 1.78449513312604381026496152832, 2.34924541118969268871658147381, 2.51449263104060758167431677344, 2.62623386413265361043416781078, 3.02860342749846366064179571852, 3.10199337933578825288728933139, 3.50464013593630016629411799984, 3.89076450321693730876667080315, 3.94653290967398345098659463652, 4.44082368979591677309874429857, 4.56650918508809601958237593532, 4.98184243954086951435181798117, 5.01503037967307435874493270303, 5.15497109352716466242633866706, 5.43590190153375436318456036634, 6.02703724300821650857947394985, 6.20193679202515745442359201255, 6.24219707712050805445023914349, 6.65067766531095354132003118116, 7.08232546690759435107189515270, 7.11733374346453031684230475801, 7.41384087811808677296044834966
