| L(s) = 1 | − 7-s + 2·9-s + 11-s + 13-s + 19-s − 23-s + 37-s − 41-s − 47-s + 53-s + 59-s − 2·63-s − 77-s + 3·81-s − 89-s − 91-s + 2·99-s − 103-s + 2·117-s + 127-s + 131-s − 133-s + 137-s + 139-s + 143-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 7-s + 2·9-s + 11-s + 13-s + 19-s − 23-s + 37-s − 41-s − 47-s + 53-s + 59-s − 2·63-s − 77-s + 3·81-s − 89-s − 91-s + 2·99-s − 103-s + 2·117-s + 127-s + 131-s − 133-s + 137-s + 139-s + 143-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16000000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16000000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.915339777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.915339777\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 23 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 41 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 53 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 59 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.752024304406931343688087219878, −8.470259163068633184176072872077, −8.104603832757456008671334297611, −7.64796911616419341403190783602, −7.17207274093829513784128685220, −7.05210573781778434121091423480, −6.57440638480371128115291869599, −6.40197976500528094166704712810, −5.88413258055005476587024174285, −5.63919416550180467860781584161, −4.95219151167126215652308984753, −4.62794526465199548114984777823, −4.06911110695905757244239574780, −3.92073907315992392663511314950, −3.45457170588125784975442313407, −3.17289441575063098489022149902, −2.39427068704309268499552587773, −1.82373677731878841011899288934, −1.35371354344164158694770612844, −0.887985730505008628732700649258,
0.887985730505008628732700649258, 1.35371354344164158694770612844, 1.82373677731878841011899288934, 2.39427068704309268499552587773, 3.17289441575063098489022149902, 3.45457170588125784975442313407, 3.92073907315992392663511314950, 4.06911110695905757244239574780, 4.62794526465199548114984777823, 4.95219151167126215652308984753, 5.63919416550180467860781584161, 5.88413258055005476587024174285, 6.40197976500528094166704712810, 6.57440638480371128115291869599, 7.05210573781778434121091423480, 7.17207274093829513784128685220, 7.64796911616419341403190783602, 8.104603832757456008671334297611, 8.470259163068633184176072872077, 8.752024304406931343688087219878
