| L(s) = 1 | + 2·4-s + 8·7-s − 4·13-s − 24·19-s − 2·25-s + 16·28-s + 18·31-s − 10·37-s + 34·49-s − 8·52-s − 10·61-s − 8·64-s − 18·67-s + 8·73-s − 48·76-s + 54·79-s − 32·91-s + 20·97-s − 4·100-s − 66·103-s − 10·109-s + 16·121-s + 36·124-s + 127-s + 131-s − 192·133-s + 137-s + ⋯ |
| L(s) = 1 | + 4-s + 3.02·7-s − 1.10·13-s − 5.50·19-s − 2/5·25-s + 3.02·28-s + 3.23·31-s − 1.64·37-s + 34/7·49-s − 1.10·52-s − 1.28·61-s − 64-s − 2.19·67-s + 0.936·73-s − 5.50·76-s + 6.07·79-s − 3.35·91-s + 2.03·97-s − 2/5·100-s − 6.50·103-s − 0.957·109-s + 1.45·121-s + 3.23·124-s + 0.0887·127-s + 0.0873·131-s − 16.6·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.340653820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.340653820\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 16 T^{2} + 135 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 32 T^{2} + 735 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 40 T^{2} + 1071 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 40 T^{2} - 609 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 8 T^{2} - 2745 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 64 T^{2} + 615 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 9 T + 94 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 27 T + 322 T^{2} - 27 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 22 T^{2} - 7437 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
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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
−7.41495048524130528321451149864, −7.37194429315091087273786319201, −6.85989308012128109816603319964, −6.69780343680384688445501847075, −6.41858843830476645926202456543, −6.40737324513263614389378452278, −6.10135545036570177256444295609, −6.05781893055791417857907078663, −5.34704237082687644252322526549, −5.28910024430568593916717361355, −4.82747982749890365010129406874, −4.78169438523338396859991499185, −4.66363814745521245174760764577, −4.37641131665095971290970142678, −4.15915654330043235504265386105, −3.75970529978984459081468942899, −3.70103048999557360350019652362, −2.80430896274493800829682705507, −2.70691930239356925345995585184, −2.20622813798906800871787650865, −2.20251551888517151591779252145, −2.08954098240571840670828489821, −1.44429986447717910854686598759, −1.39801941983347366595440800093, −0.34629666063667590324780779153,
0.34629666063667590324780779153, 1.39801941983347366595440800093, 1.44429986447717910854686598759, 2.08954098240571840670828489821, 2.20251551888517151591779252145, 2.20622813798906800871787650865, 2.70691930239356925345995585184, 2.80430896274493800829682705507, 3.70103048999557360350019652362, 3.75970529978984459081468942899, 4.15915654330043235504265386105, 4.37641131665095971290970142678, 4.66363814745521245174760764577, 4.78169438523338396859991499185, 4.82747982749890365010129406874, 5.28910024430568593916717361355, 5.34704237082687644252322526549, 6.05781893055791417857907078663, 6.10135545036570177256444295609, 6.40737324513263614389378452278, 6.41858843830476645926202456543, 6.69780343680384688445501847075, 6.85989308012128109816603319964, 7.37194429315091087273786319201, 7.41495048524130528321451149864
