| L(s) = 1 | + 0.667·3-s + 0.307·5-s − 4.25·7-s − 2.55·9-s − 0.721·11-s − 4.50·13-s + 0.204·15-s + 7.96·17-s + 6.29·19-s − 2.83·21-s + 4.88·23-s − 4.90·25-s − 3.70·27-s + 7.38·29-s + 8.57·31-s − 0.481·33-s − 1.30·35-s − 0.903·37-s − 3.00·39-s − 9.48·41-s + 1.86·43-s − 0.784·45-s + 0.432·47-s + 11.0·49-s + 5.31·51-s − 13.1·53-s − 0.221·55-s + ⋯ |
| L(s) = 1 | + 0.385·3-s + 0.137·5-s − 1.60·7-s − 0.851·9-s − 0.217·11-s − 1.25·13-s + 0.0529·15-s + 1.93·17-s + 1.44·19-s − 0.618·21-s + 1.01·23-s − 0.981·25-s − 0.713·27-s + 1.37·29-s + 1.53·31-s − 0.0837·33-s − 0.220·35-s − 0.148·37-s − 0.481·39-s − 1.48·41-s + 0.283·43-s − 0.116·45-s + 0.0631·47-s + 1.58·49-s + 0.744·51-s − 1.80·53-s − 0.0298·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 0.667T + 3T^{2} \) |
| 5 | \( 1 - 0.307T + 5T^{2} \) |
| 7 | \( 1 + 4.25T + 7T^{2} \) |
| 11 | \( 1 + 0.721T + 11T^{2} \) |
| 13 | \( 1 + 4.50T + 13T^{2} \) |
| 17 | \( 1 - 7.96T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 - 4.88T + 23T^{2} \) |
| 29 | \( 1 - 7.38T + 29T^{2} \) |
| 31 | \( 1 - 8.57T + 31T^{2} \) |
| 37 | \( 1 + 0.903T + 37T^{2} \) |
| 41 | \( 1 + 9.48T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 - 0.432T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 3.80T + 59T^{2} \) |
| 61 | \( 1 + 6.98T + 61T^{2} \) |
| 67 | \( 1 + 4.59T + 67T^{2} \) |
| 71 | \( 1 + 1.98T + 71T^{2} \) |
| 73 | \( 1 - 3.07T + 73T^{2} \) |
| 79 | \( 1 + 1.39T + 79T^{2} \) |
| 83 | \( 1 - 8.69T + 83T^{2} \) |
| 89 | \( 1 - 3.04T + 89T^{2} \) |
| 97 | \( 1 + 5.67T + 97T^{2} \) |
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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
−7.68026510938078090119921050023, −6.74306933719791183956611373636, −6.15175149318414974576392372350, −5.39255738933506810878863003793, −4.84141229490105014301087632971, −3.49033143971759427082820711348, −3.05725533991091081439275432343, −2.65457374092320846577023622419, −1.15155196753552591261035668284, 0,
1.15155196753552591261035668284, 2.65457374092320846577023622419, 3.05725533991091081439275432343, 3.49033143971759427082820711348, 4.84141229490105014301087632971, 5.39255738933506810878863003793, 6.15175149318414974576392372350, 6.74306933719791183956611373636, 7.68026510938078090119921050023
