| L(s) = 1 | − 1.22·3-s + 0.941·5-s + 1.12·7-s − 1.50·9-s + 3.62·11-s + 4.58·13-s − 1.15·15-s − 5.35·17-s + 6.84·19-s − 1.38·21-s + 1.89·23-s − 4.11·25-s + 5.50·27-s + 0.925·29-s + 2.94·31-s − 4.43·33-s + 1.06·35-s + 8.21·37-s − 5.60·39-s − 1.69·41-s + 4.48·43-s − 1.41·45-s − 7.18·47-s − 5.72·49-s + 6.54·51-s + 6.78·53-s + 3.41·55-s + ⋯ |
| L(s) = 1 | − 0.705·3-s + 0.421·5-s + 0.426·7-s − 0.501·9-s + 1.09·11-s + 1.27·13-s − 0.297·15-s − 1.29·17-s + 1.57·19-s − 0.301·21-s + 0.394·23-s − 0.822·25-s + 1.06·27-s + 0.171·29-s + 0.528·31-s − 0.772·33-s + 0.179·35-s + 1.35·37-s − 0.898·39-s − 0.264·41-s + 0.684·43-s − 0.211·45-s − 1.04·47-s − 0.817·49-s + 0.916·51-s + 0.932·53-s + 0.460·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.051328428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.051328428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 1.22T + 3T^{2} \) |
| 5 | \( 1 - 0.941T + 5T^{2} \) |
| 7 | \( 1 - 1.12T + 7T^{2} \) |
| 11 | \( 1 - 3.62T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 + 5.35T + 17T^{2} \) |
| 19 | \( 1 - 6.84T + 19T^{2} \) |
| 23 | \( 1 - 1.89T + 23T^{2} \) |
| 29 | \( 1 - 0.925T + 29T^{2} \) |
| 31 | \( 1 - 2.94T + 31T^{2} \) |
| 37 | \( 1 - 8.21T + 37T^{2} \) |
| 41 | \( 1 + 1.69T + 41T^{2} \) |
| 43 | \( 1 - 4.48T + 43T^{2} \) |
| 47 | \( 1 + 7.18T + 47T^{2} \) |
| 53 | \( 1 - 6.78T + 53T^{2} \) |
| 59 | \( 1 + 3.89T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 6.25T + 71T^{2} \) |
| 73 | \( 1 - 8.98T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 6.63T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 7.37T + 97T^{2} \) |
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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
−7.971622002224596533637879544396, −6.81413438184949080849416408682, −6.44551825997039851477567920301, −5.79135908433684282138526437775, −5.16670686573527948452500865012, −4.35517978325060691714465303298, −3.59402906757246897124032825192, −2.64660057174661620821897336514, −1.55662473978115690972160017324, −0.794229644026441400673916322711,
0.794229644026441400673916322711, 1.55662473978115690972160017324, 2.64660057174661620821897336514, 3.59402906757246897124032825192, 4.35517978325060691714465303298, 5.16670686573527948452500865012, 5.79135908433684282138526437775, 6.44551825997039851477567920301, 6.81413438184949080849416408682, 7.971622002224596533637879544396
