| L(s) = 1 | − 0.137·3-s + 1.57·5-s + 4.44·7-s − 2.98·9-s − 3.56·11-s − 0.104·13-s − 0.216·15-s − 6.27·17-s + 3.22·19-s − 0.609·21-s − 0.996·23-s − 2.50·25-s + 0.820·27-s − 6.05·29-s − 0.827·31-s + 0.489·33-s + 7.01·35-s + 3.24·37-s + 0.0142·39-s − 10.9·41-s − 10.1·43-s − 4.70·45-s − 5.38·47-s + 12.7·49-s + 0.860·51-s − 2.80·53-s − 5.63·55-s + ⋯ |
| L(s) = 1 | − 0.0791·3-s + 0.706·5-s + 1.67·7-s − 0.993·9-s − 1.07·11-s − 0.0288·13-s − 0.0559·15-s − 1.52·17-s + 0.739·19-s − 0.132·21-s − 0.207·23-s − 0.500·25-s + 0.157·27-s − 1.12·29-s − 0.148·31-s + 0.0851·33-s + 1.18·35-s + 0.533·37-s + 0.00228·39-s − 1.70·41-s − 1.55·43-s − 0.701·45-s − 0.784·47-s + 1.81·49-s + 0.120·51-s − 0.384·53-s − 0.759·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 0.137T + 3T^{2} \) |
| 5 | \( 1 - 1.57T + 5T^{2} \) |
| 7 | \( 1 - 4.44T + 7T^{2} \) |
| 11 | \( 1 + 3.56T + 11T^{2} \) |
| 13 | \( 1 + 0.104T + 13T^{2} \) |
| 17 | \( 1 + 6.27T + 17T^{2} \) |
| 19 | \( 1 - 3.22T + 19T^{2} \) |
| 23 | \( 1 + 0.996T + 23T^{2} \) |
| 29 | \( 1 + 6.05T + 29T^{2} \) |
| 31 | \( 1 + 0.827T + 31T^{2} \) |
| 37 | \( 1 - 3.24T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 + 2.80T + 53T^{2} \) |
| 59 | \( 1 + 7.69T + 59T^{2} \) |
| 61 | \( 1 - 0.864T + 61T^{2} \) |
| 67 | \( 1 - 3.77T + 67T^{2} \) |
| 71 | \( 1 + 1.33T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 9.16T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 - 6.97T + 97T^{2} \) |
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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
−8.171015906967067085116578733407, −7.51193394579094998454464247668, −6.54217454089085464714283362392, −5.63201953954243191021376156940, −5.16579502025260940986812057114, −4.55338069674703364105205929085, −3.28892418106630352670757896383, −2.22955104313537763871161143494, −1.70909685587293737030646907698, 0,
1.70909685587293737030646907698, 2.22955104313537763871161143494, 3.28892418106630352670757896383, 4.55338069674703364105205929085, 5.16579502025260940986812057114, 5.63201953954243191021376156940, 6.54217454089085464714283362392, 7.51193394579094998454464247668, 8.171015906967067085116578733407
