| L(s) = 1 | − 4.35·2-s + 11.0·4-s + 7·7-s − 13.0·8-s − 43.5·11-s − 82·13-s − 30.5·14-s − 30.9·16-s + 78.4·17-s − 20·19-s + 190.·22-s + 130.·23-s + 357.·26-s + 77.0·28-s + 244.·29-s + 156·31-s + 239.·32-s − 342.·34-s − 186·37-s + 87.1·38-s − 165.·41-s − 164·43-s − 479.·44-s − 570·46-s − 470.·47-s + 49·49-s − 902.·52-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.37·4-s + 0.377·7-s − 0.577·8-s − 1.19·11-s − 1.74·13-s − 0.582·14-s − 0.484·16-s + 1.11·17-s − 0.241·19-s + 1.84·22-s + 1.18·23-s + 2.69·26-s + 0.519·28-s + 1.56·29-s + 0.903·31-s + 1.32·32-s − 1.72·34-s − 0.826·37-s + 0.372·38-s − 0.630·41-s − 0.581·43-s − 1.64·44-s − 1.82·46-s − 1.46·47-s + 0.142·49-s − 2.40·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 4.35T + 8T^{2} \) |
| 11 | \( 1 + 43.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 82T + 2.19e3T^{2} \) |
| 17 | \( 1 - 78.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20T + 6.85e3T^{2} \) |
| 23 | \( 1 - 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 244.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 156T + 2.97e4T^{2} \) |
| 37 | \( 1 + 186T + 5.06e4T^{2} \) |
| 41 | \( 1 + 165.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 164T + 7.95e4T^{2} \) |
| 47 | \( 1 + 470.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 156.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 156.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 790T + 2.26e5T^{2} \) |
| 67 | \( 1 - 44T + 3.00e5T^{2} \) |
| 71 | \( 1 - 444.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 126T + 3.89e5T^{2} \) |
| 79 | \( 1 + 712T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.46e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.45e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 798T + 9.12e5T^{2} \) |
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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
−8.451197787958746980682123165043, −8.131770128728905841801201329873, −7.28686349679267750504673946192, −6.72696842116784968926156911817, −5.23923926429886942539645962710, −4.77043998970589015163552518764, −3.01070291503872173611871748712, −2.24184616499905154903230776304, −1.02522391810165509156188322868, 0,
1.02522391810165509156188322868, 2.24184616499905154903230776304, 3.01070291503872173611871748712, 4.77043998970589015163552518764, 5.23923926429886942539645962710, 6.72696842116784968926156911817, 7.28686349679267750504673946192, 8.131770128728905841801201329873, 8.451197787958746980682123165043
