| L(s) = 1 | − 2·5-s − 2·7-s − 2·13-s − 2·17-s + 8·19-s − 10·23-s − 25-s + 4·35-s − 10·37-s − 12·41-s + 6·43-s − 14·47-s + 2·49-s + 2·53-s − 8·59-s + 4·61-s + 4·65-s − 14·67-s + 18·73-s − 16·79-s + 10·83-s + 4·85-s + 4·91-s − 16·95-s − 6·97-s − 12·101-s − 6·103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 0.554·13-s − 0.485·17-s + 1.83·19-s − 2.08·23-s − 1/5·25-s + 0.676·35-s − 1.64·37-s − 1.87·41-s + 0.914·43-s − 2.04·47-s + 2/7·49-s + 0.274·53-s − 1.04·59-s + 0.512·61-s + 0.496·65-s − 1.71·67-s + 2.10·73-s − 1.80·79-s + 1.09·83-s + 0.433·85-s + 0.419·91-s − 1.64·95-s − 0.609·97-s − 1.19·101-s − 0.591·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2951493189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2951493189\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.869073141952054986676514834988, −9.406806471027853192097028606626, −8.982057308331074596731283697574, −8.364648117526184691794388783482, −8.173110521573455575441686732743, −7.66936865878311624725629441079, −7.42440428137278816054368060778, −6.80957139102874314198658143881, −6.67619422419343276746333024759, −6.01032625509004315312111165013, −5.63581219000486920348424013005, −5.07201701853113120937285396111, −4.76682241780314307731601842257, −4.04282229855996479363070751214, −3.72118649948981497388303662435, −3.27878314372279031627482886553, −2.83197027244221292361290427993, −2.02072749519473538072721194102, −1.46011209873107107542489244705, −0.21634035022507637690139021622,
0.21634035022507637690139021622, 1.46011209873107107542489244705, 2.02072749519473538072721194102, 2.83197027244221292361290427993, 3.27878314372279031627482886553, 3.72118649948981497388303662435, 4.04282229855996479363070751214, 4.76682241780314307731601842257, 5.07201701853113120937285396111, 5.63581219000486920348424013005, 6.01032625509004315312111165013, 6.67619422419343276746333024759, 6.80957139102874314198658143881, 7.42440428137278816054368060778, 7.66936865878311624725629441079, 8.173110521573455575441686732743, 8.364648117526184691794388783482, 8.982057308331074596731283697574, 9.406806471027853192097028606626, 9.869073141952054986676514834988
