L(s) = 1 | + 8·9-s + 48·41-s − 8·49-s + 30·81-s + 24·89-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
L(s) = 1 | + 8/3·9-s + 7.49·41-s − 8/7·49-s + 10/3·81-s + 2.54·89-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(10.49602111\) |
\(L(\frac12)\) |
\(\approx\) |
\(10.49602111\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.12505977693934690802267176816, −5.86197909919710251196310555134, −5.66269012908206579785187372778, −5.65506345717924564612781294588, −5.55258207921656896071548240671, −4.85704260264121078331203349494, −4.85377751630285356745847945586, −4.65937574001803619265214304332, −4.42383587629519815647301654860, −4.30107082651958227584619783553, −4.20799995469679379846350094327, −4.01821367015639507002774943088, −3.53110413824044632371810507149, −3.52198262449594226102679596540, −3.32575124432422205203599727838, −2.85244636023037056017606456719, −2.61432698199595849931782372763, −2.39281445748718419741787766846, −2.25117285212118282939805919152, −1.85433128560553125211803212941, −1.68515344867657993320041582016, −1.26837630764673246398586566190, −1.00821793580506145359155993651, −0.70032565796597816919323556420, −0.57913438628799109247460072118,
0.57913438628799109247460072118, 0.70032565796597816919323556420, 1.00821793580506145359155993651, 1.26837630764673246398586566190, 1.68515344867657993320041582016, 1.85433128560553125211803212941, 2.25117285212118282939805919152, 2.39281445748718419741787766846, 2.61432698199595849931782372763, 2.85244636023037056017606456719, 3.32575124432422205203599727838, 3.52198262449594226102679596540, 3.53110413824044632371810507149, 4.01821367015639507002774943088, 4.20799995469679379846350094327, 4.30107082651958227584619783553, 4.42383587629519815647301654860, 4.65937574001803619265214304332, 4.85377751630285356745847945586, 4.85704260264121078331203349494, 5.55258207921656896071548240671, 5.65506345717924564612781294588, 5.66269012908206579785187372778, 5.86197909919710251196310555134, 6.12505977693934690802267176816