| L(s) = 1 | − 4.24·2-s + 9.99·4-s − 19.4·5-s − 4.35·7-s − 8.45·8-s + 82.6·10-s + 2.86·11-s − 13.6·13-s + 18.4·14-s − 44.0·16-s − 22.2·17-s + 51.4·19-s − 194.·20-s − 12.1·22-s − 149.·23-s + 255.·25-s + 58.0·26-s − 43.4·28-s − 132.·29-s + 252.·31-s + 254.·32-s + 94.2·34-s + 84.8·35-s − 269.·37-s − 218.·38-s + 164.·40-s − 136.·41-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.24·4-s − 1.74·5-s − 0.234·7-s − 0.373·8-s + 2.61·10-s + 0.0785·11-s − 0.291·13-s + 0.352·14-s − 0.688·16-s − 0.317·17-s + 0.620·19-s − 2.17·20-s − 0.117·22-s − 1.35·23-s + 2.04·25-s + 0.437·26-s − 0.293·28-s − 0.845·29-s + 1.46·31-s + 1.40·32-s + 0.475·34-s + 0.409·35-s − 1.19·37-s − 0.930·38-s + 0.651·40-s − 0.520·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 4.24T + 8T^{2} \) |
| 5 | \( 1 + 19.4T + 125T^{2} \) |
| 7 | \( 1 + 4.35T + 343T^{2} \) |
| 11 | \( 1 - 2.86T + 1.33e3T^{2} \) |
| 13 | \( 1 + 13.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 51.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 132.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 269.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 384.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 378.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 466.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 182.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 245.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 626.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 139.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 804.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 490.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 105.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 962.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
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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
−8.338566222618650498455524010401, −7.76245422551799403750984028160, −7.16935116408221041636466428170, −6.45440381783551119580064770675, −5.03579857766245079109431911269, −4.10925380972744117832895785792, −3.29790698252031997630538932105, −2.04286600024726757396068769376, −0.75319466994166145384749770571, 0,
0.75319466994166145384749770571, 2.04286600024726757396068769376, 3.29790698252031997630538932105, 4.10925380972744117832895785792, 5.03579857766245079109431911269, 6.45440381783551119580064770675, 7.16935116408221041636466428170, 7.76245422551799403750984028160, 8.338566222618650498455524010401
