| L(s) = 1 | − 3.37·2-s + 3.37·4-s + 5·5-s + 1.62·7-s + 15.6·8-s − 16.8·10-s − 32.8·11-s − 33.0·13-s − 5.48·14-s − 79.6·16-s + 110.·17-s − 54.3·19-s + 16.8·20-s + 110.·22-s − 67.4·23-s + 25·25-s + 111.·26-s + 5.48·28-s + 274.·29-s − 6·31-s + 143.·32-s − 371.·34-s + 8.13·35-s − 347.·37-s + 183.·38-s + 78.0·40-s + 291.·41-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.421·4-s + 0.447·5-s + 0.0878·7-s + 0.689·8-s − 0.533·10-s − 0.900·11-s − 0.704·13-s − 0.104·14-s − 1.24·16-s + 1.57·17-s − 0.655·19-s + 0.188·20-s + 1.07·22-s − 0.611·23-s + 0.200·25-s + 0.839·26-s + 0.0370·28-s + 1.75·29-s − 0.0347·31-s + 0.793·32-s − 1.87·34-s + 0.0393·35-s − 1.54·37-s + 0.781·38-s + 0.308·40-s + 1.10·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 5T \) |
| good | 2 | \( 1 + 3.37T + 8T^{2} \) |
| 7 | \( 1 - 1.62T + 343T^{2} \) |
| 11 | \( 1 + 32.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 110.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 67.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 274.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 347.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 291.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 201.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 482.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 175.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 183.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 436.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 831.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 118.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 183.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 638.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.49e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 891.T + 9.12e5T^{2} \) |
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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
−10.12249416728025618057410008971, −9.666767985475558314655697679348, −8.460780380512245500372213877398, −7.896209725887110860179543142108, −6.90805832120492354225694555464, −5.58985824535337388471699439136, −4.55736913378278604035829951614, −2.80074063198763675994116546125, −1.46206459093805403087378164364, 0,
1.46206459093805403087378164364, 2.80074063198763675994116546125, 4.55736913378278604035829951614, 5.58985824535337388471699439136, 6.90805832120492354225694555464, 7.896209725887110860179543142108, 8.460780380512245500372213877398, 9.666767985475558314655697679348, 10.12249416728025618057410008971
