| L(s) = 1 | + 0.880·2-s + 3.13·3-s − 1.22·4-s − 5-s + 2.75·6-s − 1.63·7-s − 2.83·8-s + 6.81·9-s − 0.880·10-s − 4.58·11-s − 3.83·12-s − 3.46·13-s − 1.44·14-s − 3.13·15-s − 0.0504·16-s − 7.70·17-s + 5.99·18-s + 1.11·19-s + 1.22·20-s − 5.12·21-s − 4.04·22-s − 2.48·23-s − 8.89·24-s + 25-s − 3.05·26-s + 11.9·27-s + 2.00·28-s + ⋯ |
| L(s) = 1 | + 0.622·2-s + 1.80·3-s − 0.612·4-s − 0.447·5-s + 1.12·6-s − 0.618·7-s − 1.00·8-s + 2.27·9-s − 0.278·10-s − 1.38·11-s − 1.10·12-s − 0.961·13-s − 0.384·14-s − 0.808·15-s − 0.0126·16-s − 1.86·17-s + 1.41·18-s + 0.256·19-s + 0.273·20-s − 1.11·21-s − 0.861·22-s − 0.517·23-s − 1.81·24-s + 0.200·25-s − 0.598·26-s + 2.29·27-s + 0.378·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 0.880T + 2T^{2} \) |
| 3 | \( 1 - 3.13T + 3T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 + 4.58T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 7.70T + 17T^{2} \) |
| 19 | \( 1 - 1.11T + 19T^{2} \) |
| 23 | \( 1 + 2.48T + 23T^{2} \) |
| 29 | \( 1 - 2.89T + 29T^{2} \) |
| 31 | \( 1 - 2.39T + 31T^{2} \) |
| 37 | \( 1 + 7.64T + 37T^{2} \) |
| 41 | \( 1 + 7.75T + 41T^{2} \) |
| 43 | \( 1 - 2.40T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 6.39T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 + 3.90T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 5.31T + 71T^{2} \) |
| 73 | \( 1 - 7.24T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 - 8.10T + 97T^{2} \) |
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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
−8.606164647820269013014858974671, −8.231971252384572490713309022120, −7.33833852347056778174765042874, −6.58720875008723650917543770570, −5.17030963964737248997806498961, −4.48538903354527533823231861049, −3.69885482722328734102904820689, −2.86565414178581511700601442142, −2.28192118270015010117686622947, 0,
2.28192118270015010117686622947, 2.86565414178581511700601442142, 3.69885482722328734102904820689, 4.48538903354527533823231861049, 5.17030963964737248997806498961, 6.58720875008723650917543770570, 7.33833852347056778174765042874, 8.231971252384572490713309022120, 8.606164647820269013014858974671
