| L(s) = 1 | − 4·4-s − 2·7-s − 2·13-s + 12·16-s + 10·19-s + 2·25-s + 8·28-s + 2·31-s + 10·37-s + 16·43-s − 11·49-s + 8·52-s + 10·61-s − 32·64-s + 10·67-s + 22·73-s − 40·76-s − 2·79-s + 4·91-s + 10·97-s − 8·100-s − 2·103-s − 20·109-s − 24·112-s − 10·121-s − 8·124-s + 127-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.755·7-s − 0.554·13-s + 3·16-s + 2.29·19-s + 2/5·25-s + 1.51·28-s + 0.359·31-s + 1.64·37-s + 2.43·43-s − 1.57·49-s + 1.10·52-s + 1.28·61-s − 4·64-s + 1.22·67-s + 2.57·73-s − 4.58·76-s − 0.225·79-s + 0.419·91-s + 1.01·97-s − 4/5·100-s − 0.197·103-s − 1.91·109-s − 2.26·112-s − 0.909·121-s − 0.718·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.055631988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.055631988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−10.03292564047177048993688046725, −9.911809846592584450323299545132, −9.463259026207943941620278475260, −9.277917035403339643756163323383, −9.030656670531920628561741815314, −8.153523071424416519971803161812, −7.988695506949047678660919363573, −7.67444565112721543409497728603, −7.01491595581626317194093531625, −6.51178611587823318733120043472, −5.82996264208230393975028440681, −5.53415097725298244112775365184, −4.93460111281073043164564862460, −4.79618401994606835315143031039, −3.89853889980200916525998639103, −3.78038830838795663436356938691, −3.03372768606895926394732302672, −2.56505452685828550606153552642, −1.14630803571439896169326558963, −0.63929328049410835138045782888,
0.63929328049410835138045782888, 1.14630803571439896169326558963, 2.56505452685828550606153552642, 3.03372768606895926394732302672, 3.78038830838795663436356938691, 3.89853889980200916525998639103, 4.79618401994606835315143031039, 4.93460111281073043164564862460, 5.53415097725298244112775365184, 5.82996264208230393975028440681, 6.51178611587823318733120043472, 7.01491595581626317194093531625, 7.67444565112721543409497728603, 7.988695506949047678660919363573, 8.153523071424416519971803161812, 9.030656670531920628561741815314, 9.277917035403339643756163323383, 9.463259026207943941620278475260, 9.911809846592584450323299545132, 10.03292564047177048993688046725
