| L(s) = 1 | + 1.67·2-s − 2.49·3-s − 5.19·4-s − 4.17·6-s − 22.1·8-s − 20.7·9-s − 57.5·11-s + 12.9·12-s − 45.5·13-s + 4.52·16-s + 92.0·17-s − 34.8·18-s − 125.·19-s − 96.4·22-s − 158.·23-s + 55.1·24-s − 76.2·26-s + 119.·27-s − 40.1·29-s − 49.5·31-s + 184.·32-s + 143.·33-s + 154.·34-s + 107.·36-s − 231.·37-s − 209.·38-s + 113.·39-s + ⋯ |
| L(s) = 1 | + 0.592·2-s − 0.479·3-s − 0.649·4-s − 0.284·6-s − 0.976·8-s − 0.769·9-s − 1.57·11-s + 0.311·12-s − 0.971·13-s + 0.0707·16-s + 1.31·17-s − 0.455·18-s − 1.51·19-s − 0.934·22-s − 1.43·23-s + 0.468·24-s − 0.575·26-s + 0.849·27-s − 0.257·29-s − 0.287·31-s + 1.01·32-s + 0.757·33-s + 0.777·34-s + 0.499·36-s − 1.02·37-s − 0.895·38-s + 0.466·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3224050172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3224050172\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.67T + 8T^{2} \) |
| 3 | \( 1 + 2.49T + 27T^{2} \) |
| 11 | \( 1 + 57.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 92.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 40.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 49.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 231.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 169.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 67.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 268.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 240.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 90.4T + 2.26e5T^{2} \) |
| 67 | \( 1 - 406.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 330.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 546.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 25.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 376.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 942.T + 9.12e5T^{2} \) |
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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
−9.472564384214745581715280217288, −8.294363256387160047071417790549, −7.959101046073313593011706724140, −6.63046174900242446271869027851, −5.53049425831180578764257691099, −5.35278976499750919527483616852, −4.31511017586325573387597936462, −3.22794930098681966312693515224, −2.26388737878839478153962047235, −0.25334724913389226126448741969,
0.25334724913389226126448741969, 2.26388737878839478153962047235, 3.22794930098681966312693515224, 4.31511017586325573387597936462, 5.35278976499750919527483616852, 5.53049425831180578764257691099, 6.63046174900242446271869027851, 7.959101046073313593011706724140, 8.294363256387160047071417790549, 9.472564384214745581715280217288
