| L(s) = 1 | + 3·3-s + 2·4-s + 5-s + 3·9-s + 11-s + 6·12-s − 8·13-s + 3·15-s − 2·17-s + 6·19-s + 2·20-s + 5·23-s + 5·25-s + 20·29-s − 31-s + 3·33-s + 6·36-s + 5·37-s − 24·39-s − 4·41-s − 16·43-s + 2·44-s + 3·45-s − 8·47-s − 6·51-s − 16·52-s + 6·53-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 4-s + 0.447·5-s + 9-s + 0.301·11-s + 1.73·12-s − 2.21·13-s + 0.774·15-s − 0.485·17-s + 1.37·19-s + 0.447·20-s + 1.04·23-s + 25-s + 3.71·29-s − 0.179·31-s + 0.522·33-s + 36-s + 0.821·37-s − 3.84·39-s − 0.624·41-s − 2.43·43-s + 0.301·44-s + 0.447·45-s − 1.16·47-s − 0.840·51-s − 2.21·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.387971755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.387971755\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 5 T + 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
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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
−10.96379487214046019153893921882, −10.38002010556796977932446041898, −10.06986691887802758286254371549, −9.737992586156509145277822006447, −9.234550385089923451083999202315, −8.711758289428788578337926299125, −8.575515160602076436545018895227, −7.74027183513698377277798092267, −7.70013696910997906183928782221, −6.82264597078474354697976335360, −6.74715643226183245994784074643, −6.36229882575383413690779574767, −5.16156503470598894836084540918, −4.95940787832346965196735567374, −4.52780090684663945905550295322, −3.19063344524637021861846921416, −3.16751200351392269646216578513, −2.52435428238875203424119709899, −2.25429689320404033145529027823, −1.20444238805381176355335625363,
1.20444238805381176355335625363, 2.25429689320404033145529027823, 2.52435428238875203424119709899, 3.16751200351392269646216578513, 3.19063344524637021861846921416, 4.52780090684663945905550295322, 4.95940787832346965196735567374, 5.16156503470598894836084540918, 6.36229882575383413690779574767, 6.74715643226183245994784074643, 6.82264597078474354697976335360, 7.70013696910997906183928782221, 7.74027183513698377277798092267, 8.575515160602076436545018895227, 8.711758289428788578337926299125, 9.234550385089923451083999202315, 9.737992586156509145277822006447, 10.06986691887802758286254371549, 10.38002010556796977932446041898, 10.96379487214046019153893921882
