| L(s) = 1 | + 0.364·2-s − 7.86·4-s − 5.79·5-s + 11.9·7-s − 5.78·8-s − 2.11·10-s − 48.8·11-s − 14.3·13-s + 4.36·14-s + 60.8·16-s − 36.0·17-s + 25.4·19-s + 45.5·20-s − 17.8·22-s + 206.·23-s − 91.4·25-s − 5.24·26-s − 94.2·28-s + 54.2·29-s + 151.·31-s + 68.4·32-s − 13.1·34-s − 69.4·35-s + 20.8·37-s + 9.29·38-s + 33.5·40-s + 32.8·41-s + ⋯ |
| L(s) = 1 | + 0.128·2-s − 0.983·4-s − 0.518·5-s + 0.647·7-s − 0.255·8-s − 0.0668·10-s − 1.33·11-s − 0.306·13-s + 0.0833·14-s + 0.950·16-s − 0.513·17-s + 0.307·19-s + 0.509·20-s − 0.172·22-s + 1.86·23-s − 0.731·25-s − 0.0395·26-s − 0.636·28-s + 0.347·29-s + 0.875·31-s + 0.378·32-s − 0.0662·34-s − 0.335·35-s + 0.0927·37-s + 0.0396·38-s + 0.132·40-s + 0.125·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + 241T \) |
| good | 2 | \( 1 - 0.364T + 8T^{2} \) |
| 5 | \( 1 + 5.79T + 125T^{2} \) |
| 7 | \( 1 - 11.9T + 343T^{2} \) |
| 11 | \( 1 + 48.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 36.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 25.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 206.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 54.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 20.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 32.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 443.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 284.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 26.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 375.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 302.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 148.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 309.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 745.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 984.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 326.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 732.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 508.T + 9.12e5T^{2} \) |
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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
−8.081048535883960550546363098977, −7.922413964155436795770585890808, −6.88312119049066393347152201119, −5.69461379569459366424703020844, −4.91435917733323968790589127463, −4.52860907577310601734737609217, −3.39446639432063412615565400933, −2.49893539357202846345919405079, −1.01136928564999749684726206151, 0,
1.01136928564999749684726206151, 2.49893539357202846345919405079, 3.39446639432063412615565400933, 4.52860907577310601734737609217, 4.91435917733323968790589127463, 5.69461379569459366424703020844, 6.88312119049066393347152201119, 7.922413964155436795770585890808, 8.081048535883960550546363098977
