| L(s) = 1 | + 0.631·2-s + 1.59·3-s − 1.60·4-s − 5-s + 1.00·6-s + 2.69·7-s − 2.27·8-s − 0.469·9-s − 0.631·10-s + 2.44·11-s − 2.54·12-s − 5.60·13-s + 1.70·14-s − 1.59·15-s + 1.76·16-s − 0.0625·17-s − 0.296·18-s − 7.38·19-s + 1.60·20-s + 4.28·21-s + 1.54·22-s − 5.03·23-s − 3.61·24-s + 25-s − 3.53·26-s − 5.51·27-s − 4.31·28-s + ⋯ |
| L(s) = 1 | + 0.446·2-s + 0.918·3-s − 0.800·4-s − 0.447·5-s + 0.410·6-s + 1.01·7-s − 0.804·8-s − 0.156·9-s − 0.199·10-s + 0.738·11-s − 0.735·12-s − 1.55·13-s + 0.454·14-s − 0.410·15-s + 0.441·16-s − 0.0151·17-s − 0.0699·18-s − 1.69·19-s + 0.358·20-s + 0.935·21-s + 0.329·22-s − 1.05·23-s − 0.738·24-s + 0.200·25-s − 0.693·26-s − 1.06·27-s − 0.815·28-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 | \( 1 - 0.631T + 2T^{2} \) |
| 3 | \( 1 - 1.59T + 3T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 5.60T + 13T^{2} \) |
| 17 | \( 1 + 0.0625T + 17T^{2} \) |
| 19 | \( 1 + 7.38T + 19T^{2} \) |
| 23 | \( 1 + 5.03T + 23T^{2} \) |
| 29 | \( 1 + 6.74T + 29T^{2} \) |
| 31 | \( 1 - 3.61T + 31T^{2} \) |
| 37 | \( 1 - 4.27T + 37T^{2} \) |
| 41 | \( 1 + 5.37T + 41T^{2} \) |
| 43 | \( 1 - 8.84T + 43T^{2} \) |
| 47 | \( 1 - 1.49T + 47T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 - 5.79T + 59T^{2} \) |
| 61 | \( 1 + 8.74T + 61T^{2} \) |
| 67 | \( 1 + 6.14T + 67T^{2} \) |
| 71 | \( 1 + 0.885T + 71T^{2} \) |
| 73 | \( 1 + 7.91T + 73T^{2} \) |
| 79 | \( 1 - 0.0785T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 - 0.557T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−8.732102248006452169530774675864, −8.050393197332439081757997443378, −7.53750924733144654551802159751, −6.29690634871770522922468905318, −5.34930108876195247615735793546, −4.36328877265131234987541867605, −4.06676222155034678206714924890, −2.85230810340254199178294120048, −1.91223639962220142368164556704, 0,
1.91223639962220142368164556704, 2.85230810340254199178294120048, 4.06676222155034678206714924890, 4.36328877265131234987541867605, 5.34930108876195247615735793546, 6.29690634871770522922468905318, 7.53750924733144654551802159751, 8.050393197332439081757997443378, 8.732102248006452169530774675864
