| L(s) = 1 | − 2·5-s + 5·9-s + 4·11-s + 6·13-s + 6·17-s + 12·23-s − 7·25-s − 6·37-s − 10·45-s + 5·49-s − 8·55-s + 20·59-s − 12·65-s − 24·67-s + 16·81-s − 32·83-s − 12·85-s + 20·99-s − 8·103-s − 30·109-s − 12·113-s − 24·115-s + 30·117-s − 10·121-s + 26·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 5/3·9-s + 1.20·11-s + 1.66·13-s + 1.45·17-s + 2.50·23-s − 7/5·25-s − 0.986·37-s − 1.49·45-s + 5/7·49-s − 1.07·55-s + 2.60·59-s − 1.48·65-s − 2.93·67-s + 16/9·81-s − 3.51·83-s − 1.30·85-s + 2.01·99-s − 0.788·103-s − 2.87·109-s − 1.12·113-s − 2.23·115-s + 2.77·117-s − 0.909·121-s + 2.32·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.087569194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.087569194\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−11.51934658035853730324972508152, −11.02002109623427657805421566937, −10.45332787133852351827811142108, −10.25192866628245335004599790864, −9.499410376287534354260725032362, −9.279794348719621932061108838309, −8.618797312506838930781956045142, −8.342221936283905436847609548821, −7.55506918318717107045340125087, −7.34864537020561679716376479234, −6.80857743730259293062239842643, −6.45057159331266784874117649187, −5.60103366058374838686862810103, −5.28237839942400213115685528030, −4.22324818397657834157527136281, −4.06480225066723033621650033340, −3.61102415332154168227078522061, −2.92416774946599500699904970660, −1.35309831046559301069422652713, −1.29470269968441608072715738381,
1.29470269968441608072715738381, 1.35309831046559301069422652713, 2.92416774946599500699904970660, 3.61102415332154168227078522061, 4.06480225066723033621650033340, 4.22324818397657834157527136281, 5.28237839942400213115685528030, 5.60103366058374838686862810103, 6.45057159331266784874117649187, 6.80857743730259293062239842643, 7.34864537020561679716376479234, 7.55506918318717107045340125087, 8.342221936283905436847609548821, 8.618797312506838930781956045142, 9.279794348719621932061108838309, 9.499410376287534354260725032362, 10.25192866628245335004599790864, 10.45332787133852351827811142108, 11.02002109623427657805421566937, 11.51934658035853730324972508152
