| L(s) = 1 | + 2-s − 2.79·3-s + 4-s + 1.09·5-s − 2.79·6-s + 0.952·7-s + 8-s + 4.80·9-s + 1.09·10-s + 5.11·11-s − 2.79·12-s + 5.75·13-s + 0.952·14-s − 3.05·15-s + 16-s + 6.81·17-s + 4.80·18-s + 4.37·19-s + 1.09·20-s − 2.66·21-s + 5.11·22-s + 4.48·23-s − 2.79·24-s − 3.80·25-s + 5.75·26-s − 5.04·27-s + 0.952·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.61·3-s + 0.5·4-s + 0.488·5-s − 1.14·6-s + 0.359·7-s + 0.353·8-s + 1.60·9-s + 0.345·10-s + 1.54·11-s − 0.806·12-s + 1.59·13-s + 0.254·14-s − 0.788·15-s + 0.250·16-s + 1.65·17-s + 1.13·18-s + 1.00·19-s + 0.244·20-s − 0.580·21-s + 1.09·22-s + 0.934·23-s − 0.570·24-s − 0.761·25-s + 1.12·26-s − 0.970·27-s + 0.179·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.788230609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.788230609\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 + 2.79T + 3T^{2} \) |
| 5 | \( 1 - 1.09T + 5T^{2} \) |
| 7 | \( 1 - 0.952T + 7T^{2} \) |
| 11 | \( 1 - 5.11T + 11T^{2} \) |
| 13 | \( 1 - 5.75T + 13T^{2} \) |
| 17 | \( 1 - 6.81T + 17T^{2} \) |
| 19 | \( 1 - 4.37T + 19T^{2} \) |
| 23 | \( 1 - 4.48T + 23T^{2} \) |
| 29 | \( 1 + 0.707T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 + 0.978T + 37T^{2} \) |
| 41 | \( 1 + 3.76T + 41T^{2} \) |
| 43 | \( 1 + 0.508T + 43T^{2} \) |
| 47 | \( 1 + 4.82T + 47T^{2} \) |
| 53 | \( 1 + 8.43T + 53T^{2} \) |
| 59 | \( 1 - 1.68T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 9.56T + 71T^{2} \) |
| 73 | \( 1 + 6.62T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 9.32T + 97T^{2} \) |
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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
−8.338176510945638323864664819308, −7.40411852767903103939430867014, −6.52237574024848976233797728587, −6.19967634327780776848724855233, −5.44025425568308046518155028966, −5.01515394445941589450582112732, −3.90072669057653280767394099787, −3.34904886889948005908580799102, −1.47751495660083335663277074606, −1.15416316182592530917061088872,
1.15416316182592530917061088872, 1.47751495660083335663277074606, 3.34904886889948005908580799102, 3.90072669057653280767394099787, 5.01515394445941589450582112732, 5.44025425568308046518155028966, 6.19967634327780776848724855233, 6.52237574024848976233797728587, 7.40411852767903103939430867014, 8.338176510945638323864664819308
