L(s) = 1 | − 0.571·2-s − 1.67·4-s + 2.67·5-s − 3.67·7-s + 2.10·8-s − 1.52·10-s + 3.81·11-s − 0.143·13-s + 2.10·14-s + 2.14·16-s − 4.47·20-s − 2.18·22-s − 7.52·23-s + 2.14·25-s + 0.0823·26-s + 6.14·28-s − 5.34·29-s − 8.81·31-s − 5.42·32-s − 9.81·35-s + 37-s + 5.61·40-s + 5.34·41-s + 2.81·43-s − 6.38·44-s + 4.30·46-s + 6·47-s + ⋯ |
L(s) = 1 | − 0.404·2-s − 0.836·4-s + 1.19·5-s − 1.38·7-s + 0.742·8-s − 0.483·10-s + 1.15·11-s − 0.0399·13-s + 0.561·14-s + 0.535·16-s − 0.999·20-s − 0.465·22-s − 1.56·23-s + 0.428·25-s + 0.0161·26-s + 1.16·28-s − 0.992·29-s − 1.58·31-s − 0.959·32-s − 1.65·35-s + 0.164·37-s + 0.887·40-s + 0.834·41-s + 0.429·43-s − 0.962·44-s + 0.634·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 0.571T + 2T^{2} \) |
| 5 | \( 1 - 2.67T + 5T^{2} \) |
| 7 | \( 1 + 3.67T + 7T^{2} \) |
| 11 | \( 1 - 3.81T + 11T^{2} \) |
| 13 | \( 1 + 0.143T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 7.52T + 23T^{2} \) |
| 29 | \( 1 + 5.34T + 29T^{2} \) |
| 31 | \( 1 + 8.81T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 5.34T + 41T^{2} \) |
| 43 | \( 1 - 2.81T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 8.01T + 53T^{2} \) |
| 59 | \( 1 - 3.81T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 5.38T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 0.345T + 73T^{2} \) |
| 79 | \( 1 + 6.52T + 79T^{2} \) |
| 83 | \( 1 - 2.28T + 83T^{2} \) |
| 89 | \( 1 + 8.67T + 89T^{2} \) |
| 97 | \( 1 - 5.91T + 97T^{2} \) |
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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
−8.592983887219696154452089989273, −7.49396530746904440763361249367, −6.76825454342780826192192577190, −5.80580443649574785977328357614, −5.63437673614617317595133250661, −4.12379555688929912367195833355, −3.73038752767358613399884608806, −2.40180154624353660855947826856, −1.38076250042160071913884739728, 0,
1.38076250042160071913884739728, 2.40180154624353660855947826856, 3.73038752767358613399884608806, 4.12379555688929912367195833355, 5.63437673614617317595133250661, 5.80580443649574785977328357614, 6.76825454342780826192192577190, 7.49396530746904440763361249367, 8.592983887219696154452089989273