| L(s) = 1 | − 0.837·3-s + 3.44·5-s − 0.535·7-s − 2.29·9-s + 11-s + 5.54·13-s − 2.88·15-s + 1.49·17-s + 19-s + 0.448·21-s − 3.26·23-s + 6.90·25-s + 4.43·27-s − 3.17·29-s + 0.888·31-s − 0.837·33-s − 1.84·35-s + 11.0·37-s − 4.64·39-s − 8.44·41-s + 7.61·43-s − 7.93·45-s − 3.39·47-s − 6.71·49-s − 1.25·51-s + 7.22·53-s + 3.44·55-s + ⋯ |
| L(s) = 1 | − 0.483·3-s + 1.54·5-s − 0.202·7-s − 0.766·9-s + 0.301·11-s + 1.53·13-s − 0.745·15-s + 0.363·17-s + 0.229·19-s + 0.0979·21-s − 0.680·23-s + 1.38·25-s + 0.853·27-s − 0.589·29-s + 0.159·31-s − 0.145·33-s − 0.312·35-s + 1.81·37-s − 0.743·39-s − 1.31·41-s + 1.16·43-s − 1.18·45-s − 0.495·47-s − 0.958·49-s − 0.175·51-s + 0.992·53-s + 0.465·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.184170040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.184170040\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.837T + 3T^{2} \) |
| 5 | \( 1 - 3.44T + 5T^{2} \) |
| 7 | \( 1 + 0.535T + 7T^{2} \) |
| 13 | \( 1 - 5.54T + 13T^{2} \) |
| 17 | \( 1 - 1.49T + 17T^{2} \) |
| 23 | \( 1 + 3.26T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 - 0.888T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 8.44T + 41T^{2} \) |
| 43 | \( 1 - 7.61T + 43T^{2} \) |
| 47 | \( 1 + 3.39T + 47T^{2} \) |
| 53 | \( 1 - 7.22T + 53T^{2} \) |
| 59 | \( 1 - 1.36T + 59T^{2} \) |
| 61 | \( 1 + 2.57T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 2.66T + 71T^{2} \) |
| 73 | \( 1 - 9.82T + 73T^{2} \) |
| 79 | \( 1 - 2.17T + 79T^{2} \) |
| 83 | \( 1 + 4.71T + 83T^{2} \) |
| 89 | \( 1 + 4.66T + 89T^{2} \) |
| 97 | \( 1 - 6.93T + 97T^{2} \) |
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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
−8.680461165231348905910612418405, −8.048081680770302916603999463195, −6.81121546937292855281168785007, −6.08818154900024026661228527468, −5.86517252125032566391795984203, −5.08685110139690078924295210739, −3.90296798119748229541168374277, −2.95623861908621604452654817825, −1.94699582896425940345597968448, −0.943616483952674192835814904882,
0.943616483952674192835814904882, 1.94699582896425940345597968448, 2.95623861908621604452654817825, 3.90296798119748229541168374277, 5.08685110139690078924295210739, 5.86517252125032566391795984203, 6.08818154900024026661228527468, 6.81121546937292855281168785007, 8.048081680770302916603999463195, 8.680461165231348905910612418405
