| L(s) = 1 | + 2·5-s − 5·9-s + 2·13-s − 6·17-s − 7·25-s + 4·29-s + 10·37-s − 24·41-s − 10·45-s − 5·49-s + 8·53-s − 8·61-s + 4·65-s − 4·73-s + 16·81-s − 12·85-s + 4·89-s + 20·97-s + 8·101-s − 38·109-s − 12·113-s − 10·117-s − 18·121-s − 26·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 5/3·9-s + 0.554·13-s − 1.45·17-s − 7/5·25-s + 0.742·29-s + 1.64·37-s − 3.74·41-s − 1.49·45-s − 5/7·49-s + 1.09·53-s − 1.02·61-s + 0.496·65-s − 0.468·73-s + 16/9·81-s − 1.30·85-s + 0.423·89-s + 2.03·97-s + 0.796·101-s − 3.63·109-s − 1.12·113-s − 0.924·117-s − 1.63·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.039528016718973273192662402272, −8.552472099700084871980246984615, −8.052297265441174562544922743090, −7.71262456178893161528956236885, −6.65408062467448396251062225874, −6.48929045010611195636913139276, −6.08119338191936199705626075673, −5.41207186938958458805928008478, −5.12564549383277447468893363985, −4.31704488135266103326965750563, −3.64849442843249490400769039530, −2.91927671703962873131976341826, −2.34351544305748892558428024728, −1.62873087004997173572398631453, 0,
1.62873087004997173572398631453, 2.34351544305748892558428024728, 2.91927671703962873131976341826, 3.64849442843249490400769039530, 4.31704488135266103326965750563, 5.12564549383277447468893363985, 5.41207186938958458805928008478, 6.08119338191936199705626075673, 6.48929045010611195636913139276, 6.65408062467448396251062225874, 7.71262456178893161528956236885, 8.052297265441174562544922743090, 8.552472099700084871980246984615, 9.039528016718973273192662402272
