| L(s) = 1 | − 2-s + 1.79·3-s + 4-s − 3.42·5-s − 1.79·6-s + 3.84·7-s − 8-s + 0.220·9-s + 3.42·10-s + 2.47·11-s + 1.79·12-s + 6.12·13-s − 3.84·14-s − 6.15·15-s + 16-s − 0.734·17-s − 0.220·18-s + 6.62·19-s − 3.42·20-s + 6.90·21-s − 2.47·22-s + 3.97·23-s − 1.79·24-s + 6.75·25-s − 6.12·26-s − 4.98·27-s + 3.84·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.03·3-s + 0.5·4-s − 1.53·5-s − 0.732·6-s + 1.45·7-s − 0.353·8-s + 0.0734·9-s + 1.08·10-s + 0.746·11-s + 0.518·12-s + 1.69·13-s − 1.02·14-s − 1.58·15-s + 0.250·16-s − 0.178·17-s − 0.0519·18-s + 1.52·19-s − 0.766·20-s + 1.50·21-s − 0.527·22-s + 0.829·23-s − 0.366·24-s + 1.35·25-s − 1.20·26-s − 0.959·27-s + 0.726·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.078700834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.078700834\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 7 | \( 1 - 3.84T + 7T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 - 6.12T + 13T^{2} \) |
| 17 | \( 1 + 0.734T + 17T^{2} \) |
| 19 | \( 1 - 6.62T + 19T^{2} \) |
| 23 | \( 1 - 3.97T + 23T^{2} \) |
| 29 | \( 1 + 9.41T + 29T^{2} \) |
| 31 | \( 1 - 7.20T + 31T^{2} \) |
| 37 | \( 1 + 3.50T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 8.35T + 43T^{2} \) |
| 47 | \( 1 - 3.75T + 47T^{2} \) |
| 53 | \( 1 + 9.70T + 53T^{2} \) |
| 59 | \( 1 + 8.87T + 59T^{2} \) |
| 61 | \( 1 + 9.10T + 61T^{2} \) |
| 67 | \( 1 - 5.22T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 3.35T + 97T^{2} \) |
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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
−8.249792413233356841936644583877, −7.931374536841559192418573036668, −7.49968798200453599519945216775, −6.52643646538074695783330642690, −5.42991616657503389484351058101, −4.40600595422318089550168201738, −3.62644048285160292151849183338, −3.11716002258513871512566914156, −1.74670593103364261789818945479, −0.950451952287781702388554408295,
0.950451952287781702388554408295, 1.74670593103364261789818945479, 3.11716002258513871512566914156, 3.62644048285160292151849183338, 4.40600595422318089550168201738, 5.42991616657503389484351058101, 6.52643646538074695783330642690, 7.49968798200453599519945216775, 7.931374536841559192418573036668, 8.249792413233356841936644583877
