| L(s) = 1 | + 2·5-s + 5·7-s − 6·11-s − 6·13-s + 4·17-s + 5·19-s − 4·23-s + 5·25-s + 8·29-s − 7·31-s + 10·35-s + 9·37-s + 4·41-s − 2·43-s + 2·47-s + 18·49-s + 8·53-s − 12·55-s − 10·61-s − 12·65-s + 15·67-s + 12·71-s + 11·73-s − 30·77-s − 79-s − 12·83-s + 8·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.88·7-s − 1.80·11-s − 1.66·13-s + 0.970·17-s + 1.14·19-s − 0.834·23-s + 25-s + 1.48·29-s − 1.25·31-s + 1.69·35-s + 1.47·37-s + 0.624·41-s − 0.304·43-s + 0.291·47-s + 18/7·49-s + 1.09·53-s − 1.61·55-s − 1.28·61-s − 1.48·65-s + 1.83·67-s + 1.42·71-s + 1.28·73-s − 3.41·77-s − 0.112·79-s − 1.31·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.351545130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.351545130\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 8 T + 11 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.94494209126543165424406328632, −10.87072075275514347279479372708, −10.10634625282412151359127238594, −9.852024814198108807289053012031, −9.645974251661760253846192525462, −8.890528885536642797470228626488, −8.178880345957336238189269793949, −8.098329698175852775873073983742, −7.50970555769538460703916113559, −7.34413989000089131159784952071, −6.61226052008171189603554239903, −5.70474726903762060798757274996, −5.30595750339914328143036925768, −5.27210085663573538402497997387, −4.66164764227298881821069566165, −4.06104830099305256895123266484, −2.87446724735790936959966011728, −2.55391115072584950497414244288, −1.94480432129522434333023636780, −0.972834040942710151842585743168,
0.972834040942710151842585743168, 1.94480432129522434333023636780, 2.55391115072584950497414244288, 2.87446724735790936959966011728, 4.06104830099305256895123266484, 4.66164764227298881821069566165, 5.27210085663573538402497997387, 5.30595750339914328143036925768, 5.70474726903762060798757274996, 6.61226052008171189603554239903, 7.34413989000089131159784952071, 7.50970555769538460703916113559, 8.098329698175852775873073983742, 8.178880345957336238189269793949, 8.890528885536642797470228626488, 9.645974251661760253846192525462, 9.852024814198108807289053012031, 10.10634625282412151359127238594, 10.87072075275514347279479372708, 10.94494209126543165424406328632
