| L(s) = 1 | − 3-s − 2·9-s − 4·11-s + 8·17-s + 4·19-s + 2·25-s + 2·27-s + 4·33-s − 43-s + 2·49-s − 8·51-s − 4·57-s + 12·59-s − 12·67-s + 12·73-s − 2·75-s + 7·81-s + 20·89-s + 12·97-s + 8·99-s + 16·107-s + 4·113-s − 6·121-s + 127-s + 129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 1.20·11-s + 1.94·17-s + 0.917·19-s + 2/5·25-s + 0.384·27-s + 0.696·33-s − 0.152·43-s + 2/7·49-s − 1.12·51-s − 0.529·57-s + 1.56·59-s − 1.46·67-s + 1.40·73-s − 0.230·75-s + 7/9·81-s + 2.11·89-s + 1.21·97-s + 0.804·99-s + 1.54·107-s + 0.376·113-s − 0.545·121-s + 0.0887·127-s + 0.0880·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66048 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66048 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.019316589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019316589\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−10.06461556865630516510344343556, −9.459694419062171860116050133360, −8.871443697360205618830353984672, −8.293211751235953679921282732490, −7.71545675355487479415174386910, −7.52893439973954717943235960133, −6.76032739725538633360218607561, −6.01623173019841751934749573153, −5.58320709886979424438397077414, −5.22377245095964090714622061136, −4.67054533507341595405631940127, −3.51771673502652456581889441047, −3.15689241492309320937658654565, −2.24640137767395484885426138339, −0.853389605017193579550085025657,
0.853389605017193579550085025657, 2.24640137767395484885426138339, 3.15689241492309320937658654565, 3.51771673502652456581889441047, 4.67054533507341595405631940127, 5.22377245095964090714622061136, 5.58320709886979424438397077414, 6.01623173019841751934749573153, 6.76032739725538633360218607561, 7.52893439973954717943235960133, 7.71545675355487479415174386910, 8.293211751235953679921282732490, 8.871443697360205618830353984672, 9.459694419062171860116050133360, 10.06461556865630516510344343556
