L(s) = 1 | + 2-s + 2·9-s − 11-s + 13-s + 2·18-s − 19-s − 22-s + 23-s + 26-s − 31-s − 32-s − 38-s + 46-s + 2·49-s − 62-s − 64-s + 67-s + 73-s + 2·79-s + 3·81-s − 4·83-s − 89-s + 97-s + 2·98-s − 2·99-s − 101-s + 2·117-s + ⋯ |
L(s) = 1 | + 2-s + 2·9-s − 11-s + 13-s + 2·18-s − 19-s − 22-s + 23-s + 26-s − 31-s − 32-s − 38-s + 46-s + 2·49-s − 62-s − 64-s + 67-s + 73-s + 2·79-s + 3·81-s − 4·83-s − 89-s + 97-s + 2·98-s − 2·99-s − 101-s + 2·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.212601658\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.212601658\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | | \( 1 \) |
| 79 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 83 | $C_1$ | \( ( 1 + T )^{4} \) |
| 89 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 97 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
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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
−9.451284998694393232359470644793, −9.270511619461808558546992796173, −8.749909487201758827769763861166, −8.398721715430354508987961026901, −7.909319299788355742157170532638, −7.51867421900615507262213549865, −7.09210009019073384321918082731, −6.84632698003860072509105193584, −6.46906154828270222925825075247, −5.78634651966024102463650429748, −5.43088637505186378559955317445, −5.13154463243713625867282813084, −4.48274635491993578970383196083, −4.38654243336865782138705188928, −3.73063165699825637346463930729, −3.72446257491871022433055357585, −2.84750919889196819187870037443, −2.22913284343448781901232224562, −1.70031292932922563188428338805, −1.02317678223516307668602072964,
1.02317678223516307668602072964, 1.70031292932922563188428338805, 2.22913284343448781901232224562, 2.84750919889196819187870037443, 3.72446257491871022433055357585, 3.73063165699825637346463930729, 4.38654243336865782138705188928, 4.48274635491993578970383196083, 5.13154463243713625867282813084, 5.43088637505186378559955317445, 5.78634651966024102463650429748, 6.46906154828270222925825075247, 6.84632698003860072509105193584, 7.09210009019073384321918082731, 7.51867421900615507262213549865, 7.909319299788355742157170532638, 8.398721715430354508987961026901, 8.749909487201758827769763861166, 9.270511619461808558546992796173, 9.451284998694393232359470644793