L(s) = 1 | − 12·9-s + 20·25-s − 14·49-s + 90·81-s − 8·113-s + 12·121-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 − 240·225-s + 227-s + ⋯ |
L(s) = 1 | − 4·9-s + 4·25-s − 2·49-s + 10·81-s − 0.752·113-s + 1.09·121-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 − 16·225-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3820019647\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3820019647\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 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
−6.51124727178665106464595759833, −6.43126091711712095165456060778, −6.14990461490122909460150876560, −6.10224948132327949156146112607, −5.68170596021856925302131670135, −5.49484127351967012490516364774, −5.33402158100224379760996070251, −5.09124561002572294023542633619, −5.00962075529077404242404454878, −4.84748591031621635975139464903, −4.40840005774288851798571089458, −4.37596182529510836722824944579, −3.78242066559519555049596916387, −3.58330950402732531144692441401, −3.35340519501085861507139776405, −3.12538750554399393355334830493, −2.95001228245309081614793449953, −2.68253964550252967876753846398, −2.66059392762584145875861190976, −2.31157841765967597280528847082, −1.89310544637193959402721737601, −1.52876288927692105891439175842, −0.890599274575855586973926834529, −0.830916243853884145185439778155, −0.13921922143707051499433477077,
0.13921922143707051499433477077, 0.830916243853884145185439778155, 0.890599274575855586973926834529, 1.52876288927692105891439175842, 1.89310544637193959402721737601, 2.31157841765967597280528847082, 2.66059392762584145875861190976, 2.68253964550252967876753846398, 2.95001228245309081614793449953, 3.12538750554399393355334830493, 3.35340519501085861507139776405, 3.58330950402732531144692441401, 3.78242066559519555049596916387, 4.37596182529510836722824944579, 4.40840005774288851798571089458, 4.84748591031621635975139464903, 5.00962075529077404242404454878, 5.09124561002572294023542633619, 5.33402158100224379760996070251, 5.49484127351967012490516364774, 5.68170596021856925302131670135, 6.10224948132327949156146112607, 6.14990461490122909460150876560, 6.43126091711712095165456060778, 6.51124727178665106464595759833