| L(s) = 1 | + 1.44·3-s + 2.02·5-s − 4.89·7-s − 0.910·9-s + 2.40·11-s − 1.19·13-s + 2.92·15-s + 1.22·17-s − 3.46·19-s − 7.07·21-s − 0.910·25-s − 5.65·27-s + 8.90·29-s + 6.51·31-s + 3.48·33-s − 9.90·35-s − 9.45·37-s − 1.72·39-s − 7.19·41-s − 3.89·43-s − 1.84·45-s − 3.20·47-s + 16.9·49-s + 1.76·51-s − 5.15·53-s + 4.87·55-s − 5.01·57-s + ⋯ |
| L(s) = 1 | + 0.834·3-s + 0.904·5-s − 1.85·7-s − 0.303·9-s + 0.726·11-s − 0.330·13-s + 0.754·15-s + 0.296·17-s − 0.795·19-s − 1.54·21-s − 0.182·25-s − 1.08·27-s + 1.65·29-s + 1.17·31-s + 0.605·33-s − 1.67·35-s − 1.55·37-s − 0.275·39-s − 1.12·41-s − 0.594·43-s − 0.274·45-s − 0.467·47-s + 2.42·49-s + 0.247·51-s − 0.708·53-s + 0.656·55-s − 0.664·57-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 | \( 1 - 1.44T + 3T^{2} \) |
| 5 | \( 1 - 2.02T + 5T^{2} \) |
| 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 - 2.40T + 11T^{2} \) |
| 13 | \( 1 + 1.19T + 13T^{2} \) |
| 17 | \( 1 - 1.22T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 29 | \( 1 - 8.90T + 29T^{2} \) |
| 31 | \( 1 - 6.51T + 31T^{2} \) |
| 37 | \( 1 + 9.45T + 37T^{2} \) |
| 41 | \( 1 + 7.19T + 41T^{2} \) |
| 43 | \( 1 + 3.89T + 43T^{2} \) |
| 47 | \( 1 + 3.20T + 47T^{2} \) |
| 53 | \( 1 + 5.15T + 53T^{2} \) |
| 59 | \( 1 + 5.88T + 59T^{2} \) |
| 61 | \( 1 + 4.13T + 61T^{2} \) |
| 67 | \( 1 + 2.40T + 67T^{2} \) |
| 71 | \( 1 + 7.84T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 7.19T + 83T^{2} \) |
| 89 | \( 1 + 3.16T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−8.296231092118206277824320004476, −7.16279263060564692743452390963, −6.37069850267024475774661432622, −6.19708649216948190668137392343, −5.09985835962503029650783835051, −3.98777270109495928432074337927, −3.14112391382351171659038578408, −2.70424086963176305220121323361, −1.62026166754184965942299973128, 0,
1.62026166754184965942299973128, 2.70424086963176305220121323361, 3.14112391382351171659038578408, 3.98777270109495928432074337927, 5.09985835962503029650783835051, 6.19708649216948190668137392343, 6.37069850267024475774661432622, 7.16279263060564692743452390963, 8.296231092118206277824320004476
