| L(s) = 1 | − 3-s + 21.2·5-s − 31.6·7-s − 26·9-s + 0.712·11-s + 38.9·13-s − 21.2·15-s − 54.1·17-s + 19·19-s + 31.6·21-s + 61.7·23-s + 328.·25-s + 53·27-s + 225.·29-s − 131.·31-s − 0.712·33-s − 673.·35-s − 94.1·37-s − 38.9·39-s − 108.·41-s − 205.·43-s − 553.·45-s − 523.·47-s + 658.·49-s + 54.1·51-s − 560.·53-s + 15.1·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s + 1.90·5-s − 1.70·7-s − 0.962·9-s + 0.0195·11-s + 0.830·13-s − 0.366·15-s − 0.772·17-s + 0.229·19-s + 0.328·21-s + 0.560·23-s + 2.62·25-s + 0.377·27-s + 1.44·29-s − 0.762·31-s − 0.00375·33-s − 3.25·35-s − 0.418·37-s − 0.159·39-s − 0.414·41-s − 0.728·43-s − 1.83·45-s − 1.62·47-s + 1.91·49-s + 0.148·51-s − 1.45·53-s + 0.0371·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 + T + 27T^{2} \) |
| 5 | \( 1 - 21.2T + 125T^{2} \) |
| 7 | \( 1 + 31.6T + 343T^{2} \) |
| 11 | \( 1 - 0.712T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 54.1T + 4.91e3T^{2} \) |
| 23 | \( 1 - 61.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 225.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 131.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 94.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 108.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 205.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 523.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 560.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 498.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 179.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 985.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 904.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 724.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 208.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 683.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.128769234310573620005228056931, −8.485242741495349942554915888680, −6.75128826479355224509431381338, −6.41848853858587837272007550388, −5.79728242906429228223159347915, −4.93072113533786590667549632860, −3.28167449050424977321876635221, −2.71232307156100525535306455302, −1.45031275962286229185646455701, 0,
1.45031275962286229185646455701, 2.71232307156100525535306455302, 3.28167449050424977321876635221, 4.93072113533786590667549632860, 5.79728242906429228223159347915, 6.41848853858587837272007550388, 6.75128826479355224509431381338, 8.485242741495349942554915888680, 9.128769234310573620005228056931
