| L(s) = 1 | − 0.585·3-s − 5-s − 4.24·7-s − 2.65·9-s − 0.585·11-s + 3.82·13-s + 0.585·15-s + 6.82·17-s + 5.41·19-s + 2.48·21-s − 4·25-s + 3.31·27-s + 0.171·29-s − 8.82·31-s + 0.343·33-s + 4.24·35-s + 5.65·37-s − 2.24·39-s − 3.82·41-s + 6.48·43-s + 2.65·45-s + 2.24·47-s + 10.9·49-s − 4·51-s + 5·53-s + 0.585·55-s − 3.17·57-s + ⋯ |
| L(s) = 1 | − 0.338·3-s − 0.447·5-s − 1.60·7-s − 0.885·9-s − 0.176·11-s + 1.06·13-s + 0.151·15-s + 1.65·17-s + 1.24·19-s + 0.542·21-s − 0.800·25-s + 0.637·27-s + 0.0318·29-s − 1.58·31-s + 0.0597·33-s + 0.717·35-s + 0.929·37-s − 0.359·39-s − 0.597·41-s + 0.988·43-s + 0.396·45-s + 0.327·47-s + 1.57·49-s − 0.560·51-s + 0.686·53-s + 0.0789·55-s − 0.420·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 0.585T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 0.585T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 - 5.41T + 19T^{2} \) |
| 29 | \( 1 - 0.171T + 29T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 + 3.82T + 41T^{2} \) |
| 43 | \( 1 - 6.48T + 43T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 53 | \( 1 - 5T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 9.82T + 61T^{2} \) |
| 67 | \( 1 + 9.65T + 67T^{2} \) |
| 71 | \( 1 + 3.07T + 71T^{2} \) |
| 73 | \( 1 + 9.48T + 73T^{2} \) |
| 79 | \( 1 - 7.31T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 + 7T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−7.87669600275718346298551928544, −7.39780715956686010196059825685, −6.42222808219751697976773385774, −5.74949874238289106673965402018, −5.42671876358889434575316725103, −3.93823899031924834765176865645, −3.39848405792676489834064964485, −2.78219948919496933445156708870, −1.12595790299866945065869493777, 0,
1.12595790299866945065869493777, 2.78219948919496933445156708870, 3.39848405792676489834064964485, 3.93823899031924834765176865645, 5.42671876358889434575316725103, 5.74949874238289106673965402018, 6.42222808219751697976773385774, 7.39780715956686010196059825685, 7.87669600275718346298551928544
