| L(s) = 1 | + 3-s + 3.61·5-s − 3.10·7-s + 9-s − 3.85·11-s − 6.23·13-s + 3.61·15-s + 4.34·17-s + 3.23·19-s − 3.10·21-s − 7.61·23-s + 8.07·25-s + 27-s + 1.61·29-s − 1.27·31-s − 3.85·33-s − 11.2·35-s − 5.74·37-s − 6.23·39-s + 3.85·41-s − 0.542·43-s + 3.61·45-s − 3.29·47-s + 2.64·49-s + 4.34·51-s + 0.860·53-s − 13.9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.61·5-s − 1.17·7-s + 0.333·9-s − 1.16·11-s − 1.73·13-s + 0.933·15-s + 1.05·17-s + 0.742·19-s − 0.677·21-s − 1.58·23-s + 1.61·25-s + 0.192·27-s + 0.300·29-s − 0.228·31-s − 0.670·33-s − 1.89·35-s − 0.944·37-s − 0.998·39-s + 0.601·41-s − 0.0827·43-s + 0.538·45-s − 0.479·47-s + 0.378·49-s + 0.608·51-s + 0.118·53-s − 1.87·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 5 | \( 1 - 3.61T + 5T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 - 4.34T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 + 7.61T + 23T^{2} \) |
| 29 | \( 1 - 1.61T + 29T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 41 | \( 1 - 3.85T + 41T^{2} \) |
| 43 | \( 1 + 0.542T + 43T^{2} \) |
| 47 | \( 1 + 3.29T + 47T^{2} \) |
| 53 | \( 1 - 0.860T + 53T^{2} \) |
| 59 | \( 1 - 6.98T + 59T^{2} \) |
| 61 | \( 1 + 0.290T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 + 8.43T + 71T^{2} \) |
| 73 | \( 1 - 3.88T + 73T^{2} \) |
| 79 | \( 1 + 9.09T + 79T^{2} \) |
| 83 | \( 1 - 1.49T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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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
−7.66272918235861326723563275417, −7.04030235675260298222083407307, −6.26884225830233552302751568205, −5.44394330995479218942333225268, −5.17831794884513735142796543933, −3.89754118142167957125633150789, −2.77269339988391630751958681666, −2.61706142606901672557797244830, −1.57852094445957712212769269221, 0,
1.57852094445957712212769269221, 2.61706142606901672557797244830, 2.77269339988391630751958681666, 3.89754118142167957125633150789, 5.17831794884513735142796543933, 5.44394330995479218942333225268, 6.26884225830233552302751568205, 7.04030235675260298222083407307, 7.66272918235861326723563275417
