| L(s) = 1 | − 0.149·2-s − 2.07·3-s − 1.97·4-s + 4.08·5-s + 0.310·6-s + 5.08·7-s + 0.596·8-s + 1.29·9-s − 0.612·10-s + 6.60·11-s + 4.09·12-s − 2.12·13-s − 0.761·14-s − 8.46·15-s + 3.86·16-s + 17-s − 0.193·18-s − 4.59·19-s − 8.08·20-s − 10.5·21-s − 0.990·22-s − 5.10·23-s − 1.23·24-s + 11.6·25-s + 0.318·26-s + 3.53·27-s − 10.0·28-s + ⋯ |
| L(s) = 1 | − 0.105·2-s − 1.19·3-s − 0.988·4-s + 1.82·5-s + 0.126·6-s + 1.92·7-s + 0.210·8-s + 0.430·9-s − 0.193·10-s + 1.99·11-s + 1.18·12-s − 0.590·13-s − 0.203·14-s − 2.18·15-s + 0.966·16-s + 0.242·17-s − 0.0456·18-s − 1.05·19-s − 1.80·20-s − 2.29·21-s − 0.211·22-s − 1.06·23-s − 0.252·24-s + 2.33·25-s + 0.0625·26-s + 0.680·27-s − 1.89·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.480581640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.480581640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + 0.149T + 2T^{2} \) |
| 3 | \( 1 + 2.07T + 3T^{2} \) |
| 5 | \( 1 - 4.08T + 5T^{2} \) |
| 7 | \( 1 - 5.08T + 7T^{2} \) |
| 11 | \( 1 - 6.60T + 11T^{2} \) |
| 13 | \( 1 + 2.12T + 13T^{2} \) |
| 19 | \( 1 + 4.59T + 19T^{2} \) |
| 23 | \( 1 + 5.10T + 23T^{2} \) |
| 29 | \( 1 + 4.00T + 29T^{2} \) |
| 31 | \( 1 + 1.51T + 31T^{2} \) |
| 37 | \( 1 - 2.41T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 - 7.94T + 43T^{2} \) |
| 47 | \( 1 + 0.669T + 47T^{2} \) |
| 53 | \( 1 - 0.270T + 53T^{2} \) |
| 61 | \( 1 - 3.88T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 7.63T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 4.89T + 79T^{2} \) |
| 83 | \( 1 + 3.63T + 83T^{2} \) |
| 89 | \( 1 - 7.06T + 89T^{2} \) |
| 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
−10.04307283078926748371096915946, −9.115048408646255122403653193598, −8.660290016565860309160557572302, −7.37717514519753815640049703510, −6.09120289003508108859964263871, −5.77588021185452533456261011960, −4.81264733118670887592994601793, −4.25098791348255666430930267187, −1.95951859466030437542398096135, −1.17813631382959646944300285176,
1.17813631382959646944300285176, 1.95951859466030437542398096135, 4.25098791348255666430930267187, 4.81264733118670887592994601793, 5.77588021185452533456261011960, 6.09120289003508108859964263871, 7.37717514519753815640049703510, 8.660290016565860309160557572302, 9.115048408646255122403653193598, 10.04307283078926748371096915946
