| L(s) = 1 | + 1.33·2-s − 3-s − 0.209·4-s − 1.33·6-s + 1.27·7-s − 2.95·8-s + 9-s + 1.16·11-s + 0.209·12-s − 3.61·13-s + 1.71·14-s − 3.53·16-s + 5.35·17-s + 1.33·18-s − 7.61·19-s − 1.27·21-s + 1.55·22-s + 3.41·23-s + 2.95·24-s − 4.83·26-s − 27-s − 0.267·28-s − 3.21·29-s − 1.09·31-s + 1.17·32-s − 1.16·33-s + 7.17·34-s + ⋯ |
| L(s) = 1 | + 0.946·2-s − 0.577·3-s − 0.104·4-s − 0.546·6-s + 0.483·7-s − 1.04·8-s + 0.333·9-s + 0.351·11-s + 0.0603·12-s − 1.00·13-s + 0.457·14-s − 0.884·16-s + 1.29·17-s + 0.315·18-s − 1.74·19-s − 0.279·21-s + 0.332·22-s + 0.712·23-s + 0.603·24-s − 0.947·26-s − 0.192·27-s − 0.0505·28-s − 0.597·29-s − 0.196·31-s + 0.208·32-s − 0.202·33-s + 1.22·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.33T + 2T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 11 | \( 1 - 1.16T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 - 5.35T + 17T^{2} \) |
| 19 | \( 1 + 7.61T + 19T^{2} \) |
| 23 | \( 1 - 3.41T + 23T^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 31 | \( 1 + 1.09T + 31T^{2} \) |
| 37 | \( 1 + 7.80T + 37T^{2} \) |
| 41 | \( 1 + 3.00T + 41T^{2} \) |
| 43 | \( 1 + 3.42T + 43T^{2} \) |
| 47 | \( 1 + 9.41T + 47T^{2} \) |
| 53 | \( 1 + 7.64T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 8.78T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 4.51T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.823704278507504663550044473980, −8.031348673577827918392884180508, −6.99016042591322489904245908713, −6.28275183223450232246260701693, −5.32012189263531333891628838934, −4.89133280648231016471115972636, −4.01879701112044688376517947091, −3.08947770339262508772239584255, −1.73727295341316230826782990350, 0,
1.73727295341316230826782990350, 3.08947770339262508772239584255, 4.01879701112044688376517947091, 4.89133280648231016471115972636, 5.32012189263531333891628838934, 6.28275183223450232246260701693, 6.99016042591322489904245908713, 8.031348673577827918392884180508, 8.823704278507504663550044473980
