| L(s) = 1 | + 3-s + 5-s − 2·9-s − 11-s + 5·13-s + 15-s + 17-s − 6·19-s + 4·23-s + 25-s − 5·27-s − 3·29-s − 2·31-s − 33-s − 8·37-s + 5·39-s − 10·41-s − 2·43-s − 2·45-s + 7·47-s + 51-s + 2·53-s − 55-s − 6·57-s + 14·59-s + 8·61-s + 5·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s − 0.301·11-s + 1.38·13-s + 0.258·15-s + 0.242·17-s − 1.37·19-s + 0.834·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s − 0.359·31-s − 0.174·33-s − 1.31·37-s + 0.800·39-s − 1.56·41-s − 0.304·43-s − 0.298·45-s + 1.02·47-s + 0.140·51-s + 0.274·53-s − 0.134·55-s − 0.794·57-s + 1.82·59-s + 1.02·61-s + 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.589966207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.589966207\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.84111328938610, −15.44294938389266, −14.74255052872448, −14.43633646929976, −13.66732352243810, −13.32104034430825, −12.87533034143479, −12.11898984113810, −11.33911574236059, −10.96478338748527, −10.32773503578240, −9.762488866639082, −8.797641451133289, −8.677300299746013, −8.225009366581309, −7.217335610067090, −6.713949218706865, −5.906773958954726, −5.491843213683826, −4.700628605210703, −3.600422802017984, −3.443147259742095, −2.321431320418048, −1.846030561334898, −0.6586413986493067,
0.6586413986493067, 1.846030561334898, 2.321431320418048, 3.443147259742095, 3.600422802017984, 4.700628605210703, 5.491843213683826, 5.906773958954726, 6.713949218706865, 7.217335610067090, 8.225009366581309, 8.677300299746013, 8.797641451133289, 9.762488866639082, 10.32773503578240, 10.96478338748527, 11.33911574236059, 12.11898984113810, 12.87533034143479, 13.32104034430825, 13.66732352243810, 14.43633646929976, 14.74255052872448, 15.44294938389266, 15.84111328938610
