| L(s) = 1 | + 1.80·3-s + 0.702·5-s − 1.70·7-s + 0.247·9-s − 11-s − 6.29·13-s + 1.26·15-s − 6.35·17-s + 19-s − 3.06·21-s − 6.86·23-s − 4.50·25-s − 4.96·27-s + 1.53·29-s + 1.26·31-s − 1.80·33-s − 1.19·35-s + 11.7·37-s − 11.3·39-s + 4.31·41-s − 8.90·43-s + 0.173·45-s + 7.70·47-s − 4.10·49-s − 11.4·51-s + 3.91·53-s − 0.702·55-s + ⋯ |
| L(s) = 1 | + 1.04·3-s + 0.314·5-s − 0.643·7-s + 0.0824·9-s − 0.301·11-s − 1.74·13-s + 0.326·15-s − 1.54·17-s + 0.229·19-s − 0.669·21-s − 1.43·23-s − 0.901·25-s − 0.954·27-s + 0.284·29-s + 0.227·31-s − 0.313·33-s − 0.202·35-s + 1.92·37-s − 1.81·39-s + 0.673·41-s − 1.35·43-s + 0.0259·45-s + 1.12·47-s − 0.586·49-s − 1.60·51-s + 0.537·53-s − 0.0946·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.80T + 3T^{2} \) |
| 5 | \( 1 - 0.702T + 5T^{2} \) |
| 7 | \( 1 + 1.70T + 7T^{2} \) |
| 13 | \( 1 + 6.29T + 13T^{2} \) |
| 17 | \( 1 + 6.35T + 17T^{2} \) |
| 23 | \( 1 + 6.86T + 23T^{2} \) |
| 29 | \( 1 - 1.53T + 29T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 4.31T + 41T^{2} \) |
| 43 | \( 1 + 8.90T + 43T^{2} \) |
| 47 | \( 1 - 7.70T + 47T^{2} \) |
| 53 | \( 1 - 3.91T + 53T^{2} \) |
| 59 | \( 1 - 0.424T + 59T^{2} \) |
| 61 | \( 1 - 6.09T + 61T^{2} \) |
| 67 | \( 1 - 7.02T + 67T^{2} \) |
| 71 | \( 1 - 0.952T + 71T^{2} \) |
| 73 | \( 1 + 5.42T + 73T^{2} \) |
| 79 | \( 1 - 8.61T + 79T^{2} \) |
| 83 | \( 1 + 4.35T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 3.39T + 97T^{2} \) |
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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
−9.062319685111176035210618015313, −8.141020511548070722155044920254, −7.56214378022835661284790256954, −6.61686440717986864699352005814, −5.77721859929575623847113362134, −4.66966180372239342422548610927, −3.78469913455811738037785779733, −2.53543300183947346062022128335, −2.26880201223542499378554411159, 0,
2.26880201223542499378554411159, 2.53543300183947346062022128335, 3.78469913455811738037785779733, 4.66966180372239342422548610927, 5.77721859929575623847113362134, 6.61686440717986864699352005814, 7.56214378022835661284790256954, 8.141020511548070722155044920254, 9.062319685111176035210618015313
