| L(s) = 1 | + 2-s − 1.43·3-s + 4-s − 3.67·5-s − 1.43·6-s − 3.77·7-s + 8-s − 0.927·9-s − 3.67·10-s + 0.629·11-s − 1.43·12-s + 4.53·13-s − 3.77·14-s + 5.28·15-s + 16-s − 3.06·17-s − 0.927·18-s − 19-s − 3.67·20-s + 5.42·21-s + 0.629·22-s + 3.55·23-s − 1.43·24-s + 8.48·25-s + 4.53·26-s + 5.65·27-s − 3.77·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.831·3-s + 0.5·4-s − 1.64·5-s − 0.587·6-s − 1.42·7-s + 0.353·8-s − 0.309·9-s − 1.16·10-s + 0.189·11-s − 0.415·12-s + 1.25·13-s − 1.00·14-s + 1.36·15-s + 0.250·16-s − 0.743·17-s − 0.218·18-s − 0.229·19-s − 0.821·20-s + 1.18·21-s + 0.134·22-s + 0.741·23-s − 0.293·24-s + 1.69·25-s + 0.888·26-s + 1.08·27-s − 0.712·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 1.43T + 3T^{2} \) |
| 5 | \( 1 + 3.67T + 5T^{2} \) |
| 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 - 0.629T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 23 | \( 1 - 3.55T + 23T^{2} \) |
| 29 | \( 1 + 9.36T + 29T^{2} \) |
| 31 | \( 1 - 0.222T + 31T^{2} \) |
| 37 | \( 1 - 7.87T + 37T^{2} \) |
| 41 | \( 1 - 8.51T + 41T^{2} \) |
| 43 | \( 1 - 0.278T + 43T^{2} \) |
| 47 | \( 1 - 7.11T + 47T^{2} \) |
| 53 | \( 1 - 2.27T + 53T^{2} \) |
| 59 | \( 1 + 6.76T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 0.0542T + 71T^{2} \) |
| 73 | \( 1 - 1.62T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 0.484T + 89T^{2} \) |
| 97 | \( 1 + 1.38T + 97T^{2} \) |
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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
−7.36004421150835586032714051874, −6.52468581982403782802084274090, −6.20517663667590086546586169278, −5.48656940078736793221714961703, −4.48972342327167671713038644917, −3.91129342448394650658174838113, −3.38535232645446554325722009696, −2.60181739652752614052823107419, −0.915923967980260723896727888297, 0,
0.915923967980260723896727888297, 2.60181739652752614052823107419, 3.38535232645446554325722009696, 3.91129342448394650658174838113, 4.48972342327167671713038644917, 5.48656940078736793221714961703, 6.20517663667590086546586169278, 6.52468581982403782802084274090, 7.36004421150835586032714051874
