| L(s) = 1 | − 2-s − 2.44·3-s + 4-s + 1.36·5-s + 2.44·6-s − 7-s − 8-s + 2.98·9-s − 1.36·10-s − 4.61·11-s − 2.44·12-s − 4.84·13-s + 14-s − 3.33·15-s + 16-s − 3.79·17-s − 2.98·18-s + 3.21·19-s + 1.36·20-s + 2.44·21-s + 4.61·22-s + 4.41·23-s + 2.44·24-s − 3.14·25-s + 4.84·26-s + 0.0450·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.41·3-s + 0.5·4-s + 0.609·5-s + 0.998·6-s − 0.377·7-s − 0.353·8-s + 0.993·9-s − 0.431·10-s − 1.39·11-s − 0.706·12-s − 1.34·13-s + 0.267·14-s − 0.860·15-s + 0.250·16-s − 0.921·17-s − 0.702·18-s + 0.737·19-s + 0.304·20-s + 0.533·21-s + 0.983·22-s + 0.920·23-s + 0.499·24-s − 0.628·25-s + 0.949·26-s + 0.00866·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 - 1.36T + 5T^{2} \) |
| 11 | \( 1 + 4.61T + 11T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 19 | \( 1 - 3.21T + 19T^{2} \) |
| 23 | \( 1 - 4.41T + 23T^{2} \) |
| 29 | \( 1 + 2.55T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 - 8.82T + 37T^{2} \) |
| 41 | \( 1 - 7.26T + 41T^{2} \) |
| 43 | \( 1 - 0.909T + 43T^{2} \) |
| 47 | \( 1 - 1.93T + 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 + 6.45T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 1.03T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 9.99T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + 3.90T + 83T^{2} \) |
| 89 | \( 1 + 0.397T + 89T^{2} \) |
| 97 | \( 1 + 5.05T + 97T^{2} \) |
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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
−7.40050819065216059160436503560, −7.19025867281668347144009481398, −6.18921316177817916686455984322, −5.69251914539475717763304454125, −5.09230992979687425616242790082, −4.35590331545428073921188313880, −2.79207019531778904918331078152, −2.32381697528787198489919612308, −0.900366447700272876368445720336, 0,
0.900366447700272876368445720336, 2.32381697528787198489919612308, 2.79207019531778904918331078152, 4.35590331545428073921188313880, 5.09230992979687425616242790082, 5.69251914539475717763304454125, 6.18921316177817916686455984322, 7.19025867281668347144009481398, 7.40050819065216059160436503560
