| L(s) = 1 | + 2-s − 3.00·3-s + 4-s + 4.03·5-s − 3.00·6-s − 2.78·7-s + 8-s + 6.00·9-s + 4.03·10-s − 11-s − 3.00·12-s − 3.22·13-s − 2.78·14-s − 12.1·15-s + 16-s + 3.08·17-s + 6.00·18-s − 4.36·19-s + 4.03·20-s + 8.34·21-s − 22-s + 1.77·23-s − 3.00·24-s + 11.3·25-s − 3.22·26-s − 9.02·27-s − 2.78·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 0.5·4-s + 1.80·5-s − 1.22·6-s − 1.05·7-s + 0.353·8-s + 2.00·9-s + 1.27·10-s − 0.301·11-s − 0.866·12-s − 0.894·13-s − 0.743·14-s − 3.13·15-s + 0.250·16-s + 0.747·17-s + 1.41·18-s − 1.00·19-s + 0.903·20-s + 1.82·21-s − 0.213·22-s + 0.370·23-s − 0.612·24-s + 2.26·25-s − 0.632·26-s − 1.73·27-s − 0.525·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 3.00T + 3T^{2} \) |
| 5 | \( 1 - 4.03T + 5T^{2} \) |
| 7 | \( 1 + 2.78T + 7T^{2} \) |
| 13 | \( 1 + 3.22T + 13T^{2} \) |
| 17 | \( 1 - 3.08T + 17T^{2} \) |
| 19 | \( 1 + 4.36T + 19T^{2} \) |
| 23 | \( 1 - 1.77T + 23T^{2} \) |
| 29 | \( 1 + 7.34T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 - 0.840T + 37T^{2} \) |
| 41 | \( 1 - 2.48T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 2.30T + 47T^{2} \) |
| 53 | \( 1 + 7.93T + 53T^{2} \) |
| 59 | \( 1 + 0.0306T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 6.14T + 67T^{2} \) |
| 71 | \( 1 - 8.17T + 71T^{2} \) |
| 73 | \( 1 + 4.72T + 73T^{2} \) |
| 79 | \( 1 + 7.87T + 79T^{2} \) |
| 83 | \( 1 - 5.82T + 83T^{2} \) |
| 89 | \( 1 - 2.63T + 89T^{2} \) |
| 97 | \( 1 - 1.35T + 97T^{2} \) |
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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
−7.57100151336139279741945710567, −6.76968979916108857466781750495, −6.30911344929100697656114633156, −5.80343604206680854672838187379, −5.22235441406539858721939537741, −4.68606991064166105548542832480, −3.42084109304540864333708921681, −2.36015885080892414718125468091, −1.46327270715200065378703628336, 0,
1.46327270715200065378703628336, 2.36015885080892414718125468091, 3.42084109304540864333708921681, 4.68606991064166105548542832480, 5.22235441406539858721939537741, 5.80343604206680854672838187379, 6.30911344929100697656114633156, 6.76968979916108857466781750495, 7.57100151336139279741945710567
