L(s) = 1 | − 2·2-s − 4-s − 7-s + 8·8-s + 9-s + 8·11-s + 2·14-s − 7·16-s − 2·18-s − 16·22-s − 6·25-s + 28-s − 4·29-s − 14·32-s − 36-s + 12·37-s − 8·43-s − 8·44-s + 49-s + 12·50-s + 12·53-s − 8·56-s + 8·58-s − 63-s + 35·64-s + 8·67-s + 8·72-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1/2·4-s − 0.377·7-s + 2.82·8-s + 1/3·9-s + 2.41·11-s + 0.534·14-s − 7/4·16-s − 0.471·18-s − 3.41·22-s − 6/5·25-s + 0.188·28-s − 0.742·29-s − 2.47·32-s − 1/6·36-s + 1.97·37-s − 1.21·43-s − 1.20·44-s + 1/7·49-s + 1.69·50-s + 1.64·53-s − 1.06·56-s + 1.05·58-s − 0.125·63-s + 35/8·64-s + 0.977·67-s + 0.942·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3258344524\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3258344524\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−13.09697998821175856125894317781, −11.98659319295797721290412051260, −11.55595081754559805664353277943, −10.77789244508403227045944829665, −9.863214598051417092763914791206, −9.724661210726173973340366981146, −9.163045893798789171919999202805, −8.672365196683144779961098572501, −7.977686193886306753143677629185, −7.25047783802838427330028450541, −6.53484921474790582970192984243, −5.48510765590071063887274763802, −4.13559084050773741974089362056, −4.01671863219499060051377207353, −1.40786771277750203137263322761,
1.40786771277750203137263322761, 4.01671863219499060051377207353, 4.13559084050773741974089362056, 5.48510765590071063887274763802, 6.53484921474790582970192984243, 7.25047783802838427330028450541, 7.977686193886306753143677629185, 8.672365196683144779961098572501, 9.163045893798789171919999202805, 9.724661210726173973340366981146, 9.863214598051417092763914791206, 10.77789244508403227045944829665, 11.55595081754559805664353277943, 11.98659319295797721290412051260, 13.09697998821175856125894317781