| L(s) = 1 | − 2·3-s + 2·9-s − 8·16-s + 18·23-s + 25-s + 4·27-s − 14·37-s − 24·47-s + 16·48-s − 12·53-s + 26·67-s − 36·69-s − 12·71-s − 2·75-s − 3·81-s − 34·97-s + 8·103-s + 28·111-s − 42·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 48·141-s − 16·144-s + 149-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2/3·9-s − 2·16-s + 3.75·23-s + 1/5·25-s + 0.769·27-s − 2.30·37-s − 3.50·47-s + 2.30·48-s − 1.64·53-s + 3.17·67-s − 4.33·69-s − 1.42·71-s − 0.230·75-s − 1/3·81-s − 3.45·97-s + 0.788·103-s + 2.65·111-s − 3.95·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.04·141-s − 4/3·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3868462008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3868462008\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 - 5 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 35 T^{2} + p^{2} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 7 T + p T^{2} )^{2}( 1 - 25 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + 12 T + p T^{2} )^{2}( 1 + 50 T^{2} + p^{2} T^{4} ) \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 - 70 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2}( 1 + 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 35 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 17 T + p T^{2} )^{2}( 1 + 95 T^{2} + p^{2} T^{4} ) \) |
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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
−11.48835924044194409813873677265, −11.06498465641040350573303438256, −10.88138989192355342832196302463, −10.80504971422507305312653999049, −10.53863727231625196018524193354, −9.704271736043385903489154722322, −9.682303072155755329859385347862, −9.444252028395126395372971013627, −8.813174501455526830475471380197, −8.796465424989386929398989851816, −8.278663877808189586967777173745, −8.083045099302244928948101749412, −7.17128803580651590665461554075, −7.15190906861610606750797165916, −6.70998683208428034530949216352, −6.60354026877879547142645090208, −6.31419279479992790599764988959, −5.38481502662120442712443357971, −5.18591295389126206683905173421, −4.90586862298137220451443600131, −4.69283492377536908539427929709, −3.94953060036178666445540372141, −3.14810190884510550788584453401, −2.85991361577896401400185087078, −1.63620250690544105536133033406,
1.63620250690544105536133033406, 2.85991361577896401400185087078, 3.14810190884510550788584453401, 3.94953060036178666445540372141, 4.69283492377536908539427929709, 4.90586862298137220451443600131, 5.18591295389126206683905173421, 5.38481502662120442712443357971, 6.31419279479992790599764988959, 6.60354026877879547142645090208, 6.70998683208428034530949216352, 7.15190906861610606750797165916, 7.17128803580651590665461554075, 8.083045099302244928948101749412, 8.278663877808189586967777173745, 8.796465424989386929398989851816, 8.813174501455526830475471380197, 9.444252028395126395372971013627, 9.682303072155755329859385347862, 9.704271736043385903489154722322, 10.53863727231625196018524193354, 10.80504971422507305312653999049, 10.88138989192355342832196302463, 11.06498465641040350573303438256, 11.48835924044194409813873677265
