| L(s) = 1 | − 1.71·2-s + 3·3-s − 5.05·4-s + 14.8·5-s − 5.14·6-s + 22.4·8-s + 9·9-s − 25.4·10-s − 11·11-s − 15.1·12-s − 31.5·13-s + 44.5·15-s + 1.97·16-s − 68.3·17-s − 15.4·18-s − 113.·19-s − 75.0·20-s + 18.8·22-s − 99.7·23-s + 67.2·24-s + 95.3·25-s + 54.1·26-s + 27·27-s + 250.·29-s − 76.4·30-s + 139.·31-s − 182.·32-s + ⋯ |
| L(s) = 1 | − 0.606·2-s + 0.577·3-s − 0.631·4-s + 1.32·5-s − 0.350·6-s + 0.990·8-s + 0.333·9-s − 0.805·10-s − 0.301·11-s − 0.364·12-s − 0.672·13-s + 0.766·15-s + 0.0308·16-s − 0.974·17-s − 0.202·18-s − 1.37·19-s − 0.838·20-s + 0.182·22-s − 0.904·23-s + 0.571·24-s + 0.762·25-s + 0.408·26-s + 0.192·27-s + 1.60·29-s − 0.465·30-s + 0.807·31-s − 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.824280015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.824280015\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 1.71T + 8T^{2} \) |
| 5 | \( 1 - 14.8T + 125T^{2} \) |
| 13 | \( 1 + 31.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 68.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 99.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 250.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 45.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 381.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 282.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 170.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 94.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 163.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 864.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 382.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 705.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 626.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 551.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 868.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.43e3T + 9.12e5T^{2} \) |
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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
−8.985576556887810561788027678244, −8.509786327166703392917724928500, −7.67925317703998171546232749262, −6.65717619729423009870454295836, −5.86500763923955976995148058480, −4.76215207859508814623827404331, −4.15855584213969083729364521916, −2.55469897389398681869185621003, −2.00936452909871820037143192545, −0.68944769344244360292613163839,
0.68944769344244360292613163839, 2.00936452909871820037143192545, 2.55469897389398681869185621003, 4.15855584213969083729364521916, 4.76215207859508814623827404331, 5.86500763923955976995148058480, 6.65717619729423009870454295836, 7.67925317703998171546232749262, 8.509786327166703392917724928500, 8.985576556887810561788027678244
