| L(s) = 1 | − 2·3-s − 2·5-s + 3·9-s + 12·13-s + 4·15-s − 25-s − 4·27-s − 16·31-s + 4·37-s − 24·39-s − 4·41-s + 8·43-s − 6·45-s + 10·49-s − 12·53-s − 24·65-s + 16·67-s − 24·71-s + 2·75-s + 5·81-s − 8·83-s − 20·89-s + 32·93-s + 24·107-s − 8·111-s + 36·117-s + 18·121-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 9-s + 3.32·13-s + 1.03·15-s − 1/5·25-s − 0.769·27-s − 2.87·31-s + 0.657·37-s − 3.84·39-s − 0.624·41-s + 1.21·43-s − 0.894·45-s + 10/7·49-s − 1.64·53-s − 2.97·65-s + 1.95·67-s − 2.84·71-s + 0.230·75-s + 5/9·81-s − 0.878·83-s − 2.11·89-s + 3.31·93-s + 2.32·107-s − 0.759·111-s + 3.32·117-s + 1.63·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.238212068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.238212068\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−8.734379806350938919729633373185, −8.325806178665803861551864609891, −7.908853449102611121118775583871, −7.46483876337736422533928003126, −7.23987004492467640026617087566, −6.84720043549624350358626348772, −6.22258883538268339401687359678, −6.14007842572282967512468526645, −5.62852528967901714807388011888, −5.60358478788117515773885525188, −4.97146855756344697124598267090, −4.25972264733339860129855668421, −4.11548539221463832232546862790, −3.82186515933564960511372617941, −3.35154896961385924792478773281, −2.99222608237856889134273527132, −1.93756372283623915499700052105, −1.60187568340872147871724595077, −1.03975617128410289720254036218, −0.42137465715308899645928942561,
0.42137465715308899645928942561, 1.03975617128410289720254036218, 1.60187568340872147871724595077, 1.93756372283623915499700052105, 2.99222608237856889134273527132, 3.35154896961385924792478773281, 3.82186515933564960511372617941, 4.11548539221463832232546862790, 4.25972264733339860129855668421, 4.97146855756344697124598267090, 5.60358478788117515773885525188, 5.62852528967901714807388011888, 6.14007842572282967512468526645, 6.22258883538268339401687359678, 6.84720043549624350358626348772, 7.23987004492467640026617087566, 7.46483876337736422533928003126, 7.908853449102611121118775583871, 8.325806178665803861551864609891, 8.734379806350938919729633373185
