| L(s) = 1 | − 5.15·2-s + 18.5·4-s − 17.0·5-s − 7·7-s − 54.6·8-s + 87.7·10-s + 30.3·11-s − 89.5·13-s + 36.0·14-s + 132.·16-s − 13.4·17-s − 5.37·19-s − 316.·20-s − 156.·22-s − 167.·23-s + 164.·25-s + 461.·26-s − 130.·28-s + 135.·29-s − 18.9·31-s − 248.·32-s + 69.1·34-s + 119.·35-s + 402.·37-s + 27.7·38-s + 929.·40-s + 434.·41-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.32·4-s − 1.52·5-s − 0.377·7-s − 2.41·8-s + 2.77·10-s + 0.831·11-s − 1.91·13-s + 0.689·14-s + 2.07·16-s − 0.191·17-s − 0.0648·19-s − 3.53·20-s − 1.51·22-s − 1.51·23-s + 1.31·25-s + 3.48·26-s − 0.878·28-s + 0.864·29-s − 0.109·31-s − 1.37·32-s + 0.348·34-s + 0.575·35-s + 1.78·37-s + 0.118·38-s + 3.67·40-s + 1.65·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3230687889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3230687889\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 5.15T + 8T^{2} \) |
| 5 | \( 1 + 17.0T + 125T^{2} \) |
| 11 | \( 1 - 30.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 89.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 13.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.37T + 6.85e3T^{2} \) |
| 23 | \( 1 + 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 18.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 402.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 53.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 155.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 301.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 412.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 571.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 8.95T + 3.57e5T^{2} \) |
| 73 | \( 1 - 21.9T + 3.89e5T^{2} \) |
| 79 | \( 1 - 619.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 259.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 484.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 252.T + 9.12e5T^{2} \) |
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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
−11.93283554176482538891078344449, −10.97813737020974162238294603913, −9.894257056956434529048230190280, −9.157139369485933004936881841683, −7.947732515771880601345890325865, −7.48705033439005966092572252881, −6.42304439713843177507639568735, −4.22471802772146735291771510911, −2.54026570390734410868366538330, −0.54741944596227755502669230104,
0.54741944596227755502669230104, 2.54026570390734410868366538330, 4.22471802772146735291771510911, 6.42304439713843177507639568735, 7.48705033439005966092572252881, 7.947732515771880601345890325865, 9.157139369485933004936881841683, 9.894257056956434529048230190280, 10.97813737020974162238294603913, 11.93283554176482538891078344449
