L(s) = 1 | − 8·7-s + 2·9-s − 4·17-s + 8·23-s + 6·25-s + 12·41-s + 16·47-s + 34·49-s − 16·63-s + 24·71-s − 28·73-s + 16·79-s − 5·81-s + 4·89-s − 4·97-s − 8·103-s + 4·113-s + 32·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + ⋯ |
L(s) = 1 | − 3.02·7-s + 2/3·9-s − 0.970·17-s + 1.66·23-s + 6/5·25-s + 1.87·41-s + 2.33·47-s + 34/7·49-s − 2.01·63-s + 2.84·71-s − 3.27·73-s + 1.80·79-s − 5/9·81-s + 0.423·89-s − 0.406·97-s − 0.788·103-s + 0.376·113-s + 2.93·119-s + 1.63·121-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 − 0.646·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9434940715\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9434940715\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−12.46448586956374693128491949424, −12.01993603461779646561261081518, −11.05675814829446524247374047496, −10.82899110790509360137108028749, −10.31055245362243336087047752436, −9.776138170650733784260645661580, −9.423884709811879950950983404496, −8.870868915425085523508767953007, −8.851882489860000138251155333534, −7.56515021467950703892918779460, −7.20786807714381711280566941520, −6.66103041189581929330436903728, −6.44976683667966550556096328535, −5.83522495830210206987552686355, −5.08762159732160482445369213130, −4.21895121524273615476920088960, −3.73740289547189202194632871913, −2.90739036926013457245631156495, −2.60239244415137240943522716091, −0.77554292563920746309047607113,
0.77554292563920746309047607113, 2.60239244415137240943522716091, 2.90739036926013457245631156495, 3.73740289547189202194632871913, 4.21895121524273615476920088960, 5.08762159732160482445369213130, 5.83522495830210206987552686355, 6.44976683667966550556096328535, 6.66103041189581929330436903728, 7.20786807714381711280566941520, 7.56515021467950703892918779460, 8.851882489860000138251155333534, 8.870868915425085523508767953007, 9.423884709811879950950983404496, 9.776138170650733784260645661580, 10.31055245362243336087047752436, 10.82899110790509360137108028749, 11.05675814829446524247374047496, 12.01993603461779646561261081518, 12.46448586956374693128491949424