| L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 12·13-s + 16-s + 4·17-s − 18-s − 12·26-s − 32-s − 4·34-s + 36-s + 4·37-s + 4·41-s − 10·49-s + 12·52-s + 12·53-s + 4·61-s + 64-s + 4·68-s − 72-s − 8·73-s − 4·74-s + 81-s − 4·82-s − 20·89-s − 16·97-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1/3·9-s + 3.32·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 2.35·26-s − 0.176·32-s − 0.685·34-s + 1/6·36-s + 0.657·37-s + 0.624·41-s − 1.42·49-s + 1.66·52-s + 1.64·53-s + 0.512·61-s + 1/8·64-s + 0.485·68-s − 0.117·72-s − 0.936·73-s − 0.464·74-s + 1/9·81-s − 0.441·82-s − 2.11·89-s − 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.574994268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.574994268\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−8.876429119746077603127030187363, −8.831748546434365307733167695606, −8.236259450125909594320955817325, −7.907415924764968545249330860332, −7.34830902118093822137916374938, −6.60420630283526755468874076032, −6.39955394184406631525545593399, −5.66765515312328094809085573246, −5.54667504701134155415335928612, −4.40421151945420287303938949967, −3.85245454580381340504020564721, −3.43332661175523197849238647019, −2.66426897593599555100997020524, −1.48708143658565903074344661320, −1.09380620717066232730427473522,
1.09380620717066232730427473522, 1.48708143658565903074344661320, 2.66426897593599555100997020524, 3.43332661175523197849238647019, 3.85245454580381340504020564721, 4.40421151945420287303938949967, 5.54667504701134155415335928612, 5.66765515312328094809085573246, 6.39955394184406631525545593399, 6.60420630283526755468874076032, 7.34830902118093822137916374938, 7.907415924764968545249330860332, 8.236259450125909594320955817325, 8.831748546434365307733167695606, 8.876429119746077603127030187363
