| L(s) = 1 | + 1.93·3-s + 0.475·5-s + 0.733·9-s + 0.608·11-s − 0.258·13-s + 0.918·15-s − 6.67·17-s + 8.55·19-s − 0.540·23-s − 4.77·25-s − 4.37·27-s + 9.17·29-s + 3.19·31-s + 1.17·33-s + 2.11·37-s − 0.499·39-s + 41-s − 9.45·43-s + 0.348·45-s + 5.21·47-s − 12.8·51-s + 11.1·53-s + 0.289·55-s + 16.5·57-s + 6.80·59-s − 1.40·61-s − 0.122·65-s + ⋯ |
| L(s) = 1 | + 1.11·3-s + 0.212·5-s + 0.244·9-s + 0.183·11-s − 0.0716·13-s + 0.237·15-s − 1.61·17-s + 1.96·19-s − 0.112·23-s − 0.954·25-s − 0.842·27-s + 1.70·29-s + 0.573·31-s + 0.204·33-s + 0.347·37-s − 0.0799·39-s + 0.156·41-s − 1.44·43-s + 0.0519·45-s + 0.761·47-s − 1.80·51-s + 1.52·53-s + 0.0390·55-s + 2.19·57-s + 0.885·59-s − 0.179·61-s − 0.0152·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.207807408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.207807408\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 1.93T + 3T^{2} \) |
| 5 | \( 1 - 0.475T + 5T^{2} \) |
| 11 | \( 1 - 0.608T + 11T^{2} \) |
| 13 | \( 1 + 0.258T + 13T^{2} \) |
| 17 | \( 1 + 6.67T + 17T^{2} \) |
| 19 | \( 1 - 8.55T + 19T^{2} \) |
| 23 | \( 1 + 0.540T + 23T^{2} \) |
| 29 | \( 1 - 9.17T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 - 2.11T + 37T^{2} \) |
| 43 | \( 1 + 9.45T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 6.80T + 59T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 - 15.9T + 67T^{2} \) |
| 71 | \( 1 - 3.72T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 - 5.46T + 79T^{2} \) |
| 83 | \( 1 + 0.971T + 83T^{2} \) |
| 89 | \( 1 + 9.78T + 89T^{2} \) |
| 97 | \( 1 + 3.26T + 97T^{2} \) |
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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
−8.088073222073287825388889518741, −7.14335605780837210952323869174, −6.64548856658967409036369648681, −5.72856982067145666908278463448, −4.98612711592791477039336295472, −4.13181630701901829400927460795, −3.41520425447682631575067699896, −2.62266083903501365410132133723, −2.05064492158024392066332743278, −0.835680997794099837929831040694,
0.835680997794099837929831040694, 2.05064492158024392066332743278, 2.62266083903501365410132133723, 3.41520425447682631575067699896, 4.13181630701901829400927460795, 4.98612711592791477039336295472, 5.72856982067145666908278463448, 6.64548856658967409036369648681, 7.14335605780837210952323869174, 8.088073222073287825388889518741
