| L(s) = 1 | − 2-s − 2.07·3-s + 4-s − 1.29·5-s + 2.07·6-s + 1.20·7-s − 8-s + 1.29·9-s + 1.29·10-s + 4.05·11-s − 2.07·12-s − 1.83·13-s − 1.20·14-s + 2.67·15-s + 16-s − 4.82·17-s − 1.29·18-s + 7.30·19-s − 1.29·20-s − 2.50·21-s − 4.05·22-s + 9.07·23-s + 2.07·24-s − 3.33·25-s + 1.83·26-s + 3.52·27-s + 1.20·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.19·3-s + 0.5·4-s − 0.577·5-s + 0.846·6-s + 0.455·7-s − 0.353·8-s + 0.433·9-s + 0.408·10-s + 1.22·11-s − 0.598·12-s − 0.507·13-s − 0.322·14-s + 0.691·15-s + 0.250·16-s − 1.16·17-s − 0.306·18-s + 1.67·19-s − 0.288·20-s − 0.545·21-s − 0.864·22-s + 1.89·23-s + 0.423·24-s − 0.666·25-s + 0.358·26-s + 0.678·27-s + 0.227·28-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
| \(L(1)\) |
\(\approx\) |
\(0.7331579833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7331579833\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.07T + 3T^{2} \) |
| 5 | \( 1 + 1.29T + 5T^{2} \) |
| 7 | \( 1 - 1.20T + 7T^{2} \) |
| 11 | \( 1 - 4.05T + 11T^{2} \) |
| 13 | \( 1 + 1.83T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 7.30T + 19T^{2} \) |
| 23 | \( 1 - 9.07T + 23T^{2} \) |
| 29 | \( 1 + 2.24T + 29T^{2} \) |
| 31 | \( 1 + 3.67T + 31T^{2} \) |
| 37 | \( 1 + 2.39T + 37T^{2} \) |
| 41 | \( 1 - 8.29T + 41T^{2} \) |
| 43 | \( 1 - 2.85T + 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 5.92T + 59T^{2} \) |
| 61 | \( 1 - 7.26T + 61T^{2} \) |
| 67 | \( 1 + 8.94T + 67T^{2} \) |
| 71 | \( 1 + 3.76T + 71T^{2} \) |
| 73 | \( 1 - 4.88T + 73T^{2} \) |
| 79 | \( 1 - 6.26T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.532957824672588797502088974620, −7.49266404368923643446395098883, −7.10720841752228873137730218789, −6.36524313756238140667721351044, −5.50697108792900959088123876073, −4.84421851811001126212019366466, −3.95349289592170202707901838074, −2.87767669994801437081153730406, −1.53070906127240824750102450444, −0.61670414325482804742130757747,
0.61670414325482804742130757747, 1.53070906127240824750102450444, 2.87767669994801437081153730406, 3.95349289592170202707901838074, 4.84421851811001126212019366466, 5.50697108792900959088123876073, 6.36524313756238140667721351044, 7.10720841752228873137730218789, 7.49266404368923643446395098883, 8.532957824672588797502088974620
