| L(s) = 1 | + 2-s − 1.71·3-s + 4-s − 4.22·5-s − 1.71·6-s − 1.91·7-s + 8-s − 0.0518·9-s − 4.22·10-s + 0.342·11-s − 1.71·12-s − 1.67·13-s − 1.91·14-s + 7.26·15-s + 16-s + 2.13·17-s − 0.0518·18-s + 5.06·19-s − 4.22·20-s + 3.28·21-s + 0.342·22-s + 23-s − 1.71·24-s + 12.8·25-s − 1.67·26-s + 5.24·27-s − 1.91·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.991·3-s + 0.5·4-s − 1.89·5-s − 0.700·6-s − 0.723·7-s + 0.353·8-s − 0.0172·9-s − 1.33·10-s + 0.103·11-s − 0.495·12-s − 0.463·13-s − 0.511·14-s + 1.87·15-s + 0.250·16-s + 0.517·17-s − 0.0122·18-s + 1.16·19-s − 0.945·20-s + 0.717·21-s + 0.0730·22-s + 0.208·23-s − 0.350·24-s + 2.57·25-s − 0.327·26-s + 1.00·27-s − 0.361·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 1.71T + 3T^{2} \) |
| 5 | \( 1 + 4.22T + 5T^{2} \) |
| 7 | \( 1 + 1.91T + 7T^{2} \) |
| 11 | \( 1 - 0.342T + 11T^{2} \) |
| 13 | \( 1 + 1.67T + 13T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 19 | \( 1 - 5.06T + 19T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 + 3.68T + 31T^{2} \) |
| 37 | \( 1 - 5.22T + 37T^{2} \) |
| 41 | \( 1 - 6.30T + 41T^{2} \) |
| 43 | \( 1 + 2.69T + 43T^{2} \) |
| 47 | \( 1 + 3.66T + 47T^{2} \) |
| 53 | \( 1 + 3.80T + 53T^{2} \) |
| 59 | \( 1 - 3.76T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 - 1.50T + 67T^{2} \) |
| 71 | \( 1 - 9.75T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 9.01T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 3.91T + 89T^{2} \) |
| 97 | \( 1 + 0.619T + 97T^{2} \) |
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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
−7.62120601362583339917301792817, −6.91626084136595097934360122700, −6.32913505796965714999982815412, −5.35559417501580584196312679076, −4.96550389084010314493515923736, −3.97653288070597488169523351781, −3.47385714948191845490736298620, −2.72745642665229525448121783377, −0.965989545190015659742593461455, 0,
0.965989545190015659742593461455, 2.72745642665229525448121783377, 3.47385714948191845490736298620, 3.97653288070597488169523351781, 4.96550389084010314493515923736, 5.35559417501580584196312679076, 6.32913505796965714999982815412, 6.91626084136595097934360122700, 7.62120601362583339917301792817
