L(s) = 1 | − 2-s − 9-s + 3·13-s + 3·17-s + 18-s − 3·26-s + 3·29-s + 32-s − 3·34-s − 2·37-s + 3·41-s + 4·49-s − 2·53-s − 3·58-s + 3·61-s − 64-s + 3·73-s + 2·74-s − 3·82-s − 2·89-s + 3·97-s − 4·98-s − 2·101-s + 2·106-s + 3·109-s − 2·113-s − 3·117-s + ⋯ |
L(s) = 1 | − 2-s − 9-s + 3·13-s + 3·17-s + 18-s − 3·26-s + 3·29-s + 32-s − 3·34-s − 2·37-s + 3·41-s + 4·49-s − 2·53-s − 3·58-s + 3·61-s − 64-s + 3·73-s + 2·74-s − 3·82-s − 2·89-s + 3·97-s − 4·98-s − 2·101-s + 2·106-s + 3·109-s − 2·113-s − 3·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.186706910\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.186706910\) |
\(L(1)\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | | \( 1 \) |
good | 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−6.63291453588320583816204477210, −6.15227828822242522152735180767, −6.12904923794926219593187198082, −5.92656706472065396111437987045, −5.73440878559893036806229518181, −5.54300650437359506044810565373, −5.50362124000553685207637829266, −5.14382458355951634825515537200, −4.99060614554309244578703277541, −4.50679504468833932241103769960, −4.50581275880461499129304215114, −4.11864284953562135152126630277, −3.86946682190427220348141451577, −3.71189609442901170933963957361, −3.52906757319584919597598803973, −3.16504215310281493827556082027, −3.11328672026137483943424580955, −2.95374727689042182857232603560, −2.26680092853660939200980642338, −2.25898291840861027747833050121, −2.23444745983071046506044731110, −1.18976783125266095511919614404, −1.07378338559188825633651364852, −1.06815228181934673297886953534, −0.900456715950755990029111354991,
0.900456715950755990029111354991, 1.06815228181934673297886953534, 1.07378338559188825633651364852, 1.18976783125266095511919614404, 2.23444745983071046506044731110, 2.25898291840861027747833050121, 2.26680092853660939200980642338, 2.95374727689042182857232603560, 3.11328672026137483943424580955, 3.16504215310281493827556082027, 3.52906757319584919597598803973, 3.71189609442901170933963957361, 3.86946682190427220348141451577, 4.11864284953562135152126630277, 4.50581275880461499129304215114, 4.50679504468833932241103769960, 4.99060614554309244578703277541, 5.14382458355951634825515537200, 5.50362124000553685207637829266, 5.54300650437359506044810565373, 5.73440878559893036806229518181, 5.92656706472065396111437987045, 6.12904923794926219593187198082, 6.15227828822242522152735180767, 6.63291453588320583816204477210