| L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s − 21.3·7-s − 8·8-s − 10·10-s + 65.5·11-s − 0.271·13-s + 42.6·14-s + 16·16-s − 132.·17-s − 19·19-s + 20·20-s − 131.·22-s + 142.·23-s + 25·25-s + 0.542·26-s − 85.2·28-s + 83.9·29-s + 173.·31-s − 32·32-s + 265.·34-s − 106.·35-s − 433.·37-s + 38·38-s − 40·40-s − 113.·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 1.15·7-s − 0.353·8-s − 0.316·10-s + 1.79·11-s − 0.00578·13-s + 0.814·14-s + 0.250·16-s − 1.89·17-s − 0.229·19-s + 0.223·20-s − 1.27·22-s + 1.29·23-s + 0.200·25-s + 0.00408·26-s − 0.575·28-s + 0.537·29-s + 1.00·31-s − 0.176·32-s + 1.33·34-s − 0.514·35-s − 1.92·37-s + 0.162·38-s − 0.158·40-s − 0.431·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.391233437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.391233437\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 19 | \( 1 + 19T \) |
| good | 7 | \( 1 + 21.3T + 343T^{2} \) |
| 11 | \( 1 - 65.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.271T + 2.19e3T^{2} \) |
| 17 | \( 1 + 132.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 83.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 173.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 433.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 113.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 258.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 285.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 290.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 184.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 63.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 329.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 609.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 970.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 370.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 580.T + 9.12e5T^{2} \) |
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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
−9.120180230708873842601341547981, −8.554634214206402714563705898256, −7.13921294893525639849650840584, −6.54675945818226246311268787581, −6.26649117320151071636671618006, −4.83470728569051800782387383633, −3.80184899244593241307483016671, −2.82692408292338349052273021634, −1.74873445452585288682158526636, −0.62764595208430497827259757790,
0.62764595208430497827259757790, 1.74873445452585288682158526636, 2.82692408292338349052273021634, 3.80184899244593241307483016671, 4.83470728569051800782387383633, 6.26649117320151071636671618006, 6.54675945818226246311268787581, 7.13921294893525639849650840584, 8.554634214206402714563705898256, 9.120180230708873842601341547981
