| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s − 6·11-s + 4·13-s − 15-s + 6·17-s + 4·21-s − 6·23-s + 25-s + 27-s − 2·29-s − 6·33-s − 4·35-s + 8·37-s + 4·39-s − 10·41-s + 4·43-s − 45-s + 2·47-s + 9·49-s + 6·51-s − 10·53-s + 6·55-s − 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s − 0.258·15-s + 1.45·17-s + 0.872·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.04·33-s − 0.676·35-s + 1.31·37-s + 0.640·39-s − 1.56·41-s + 0.609·43-s − 0.149·45-s + 0.291·47-s + 9/7·49-s + 0.840·51-s − 1.37·53-s + 0.809·55-s − 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.15258429190514, −13.72246820817785, −13.26859767631396, −12.72176302330231, −12.13169904975374, −11.71633036400264, −11.10947780592996, −10.67081759582781, −10.32071266367248, −9.701996661340837, −9.035284876681174, −8.355569297906999, −8.017573153394662, −7.779391662547201, −7.449861391258018, −6.460594308981567, −5.723519432603038, −5.439167133521766, −4.664861959169167, −4.337454355642445, −3.488067594381015, −3.081120118736913, −2.298536059280395, −1.665182089533932, −1.070516243009339, 0,
1.070516243009339, 1.665182089533932, 2.298536059280395, 3.081120118736913, 3.488067594381015, 4.337454355642445, 4.664861959169167, 5.439167133521766, 5.723519432603038, 6.460594308981567, 7.449861391258018, 7.779391662547201, 8.017573153394662, 8.355569297906999, 9.035284876681174, 9.701996661340837, 10.32071266367248, 10.67081759582781, 11.10947780592996, 11.71633036400264, 12.13169904975374, 12.72176302330231, 13.26859767631396, 13.72246820817785, 14.15258429190514
