| L(s) = 1 | − 1.16e9·9-s − 1.23e11·13-s + 3.81e13·25-s − 1.56e15·37-s − 9.25e15·49-s − 2.80e17·61-s + 1.27e18·73-s + 1.35e18·81-s + 1.90e19·97-s + 8.78e19·109-s + 1.43e20·117-s − 1.22e20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.57e21·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 9-s − 3.23·13-s + 2·25-s − 1.97·37-s − 0.811·49-s − 3.07·61-s + 2.54·73-s + 81-s + 2.53·97-s + 3.87·109-s + 3.23·117-s − 2·121-s + 5.86·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(0.4434103305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4434103305\) |
| \(L(\frac{21}{2})\) |
|
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 | $C_2$ | \( 1 + p^{19} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 116377132 T + p^{19} T^{2} )( 1 + 116377132 T + p^{19} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 61912545914 T + p^{19} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 252454137272 T + p^{19} T^{2} )( 1 + 252454137272 T + p^{19} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 189181140858356 T + p^{19} T^{2} )( 1 + 189181140858356 T + p^{19} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 780590048220370 T + p^{19} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 666942065597816 T + p^{19} T^{2} )( 1 + 666942065597816 T + p^{19} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 140464646991065866 T + p^{19} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 273531389481429392 T + p^{19} T^{2} )( 1 + 273531389481429392 T + p^{19} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 639446467598381530 T + p^{19} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 1962240154233379276 T + p^{19} T^{2} )( 1 + 1962240154233379276 T + p^{19} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 9502640950925166898 T + p^{19} T^{2} )^{2} \) |
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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
−12.07083362140513615608489574850, −11.70412611675714148576697298742, −10.74873462151897805571811724331, −10.46934536049047568988867078303, −9.713718084963480912654973851793, −9.222003689425977359986856826070, −8.705196286194081168957047849511, −7.921235187372334490867976096523, −7.37682899080046907336775663191, −6.90951604931895169447090835281, −6.24056103114353748280420388372, −5.33353530570016624557191717278, −4.83093431425724251025202600394, −4.70327144242812020070645934692, −3.34388428815221452555788577001, −3.02636715138154812888545913418, −2.31658324927249649031274123336, −1.90603351061038415306272987587, −0.832942563641360724527441219197, −0.17111161521584434687931261526,
0.17111161521584434687931261526, 0.832942563641360724527441219197, 1.90603351061038415306272987587, 2.31658324927249649031274123336, 3.02636715138154812888545913418, 3.34388428815221452555788577001, 4.70327144242812020070645934692, 4.83093431425724251025202600394, 5.33353530570016624557191717278, 6.24056103114353748280420388372, 6.90951604931895169447090835281, 7.37682899080046907336775663191, 7.921235187372334490867976096523, 8.705196286194081168957047849511, 9.222003689425977359986856826070, 9.713718084963480912654973851793, 10.46934536049047568988867078303, 10.74873462151897805571811724331, 11.70412611675714148576697298742, 12.07083362140513615608489574850
