| L(s) = 1 | − 2·2-s + 8.71·3-s + 4·4-s − 10.2·5-s − 17.4·6-s − 8·8-s + 49.0·9-s + 20.4·10-s − 47.6·11-s + 34.8·12-s + 44.4·13-s − 88.9·15-s + 16·16-s − 24.4·17-s − 98.0·18-s − 116.·19-s − 40.8·20-s + 95.2·22-s + 23·23-s − 69.7·24-s − 20.9·25-s − 88.8·26-s + 192.·27-s − 57.1·29-s + 177.·30-s − 156.·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.67·3-s + 0.5·4-s − 0.912·5-s − 1.18·6-s − 0.353·8-s + 1.81·9-s + 0.645·10-s − 1.30·11-s + 0.839·12-s + 0.948·13-s − 1.53·15-s + 0.250·16-s − 0.349·17-s − 1.28·18-s − 1.40·19-s − 0.456·20-s + 0.923·22-s + 0.208·23-s − 0.593·24-s − 0.167·25-s − 0.670·26-s + 1.36·27-s − 0.366·29-s + 1.08·30-s − 0.906·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.067841560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.067841560\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 8.71T + 27T^{2} \) |
| 5 | \( 1 + 10.2T + 125T^{2} \) |
| 11 | \( 1 + 47.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 57.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 131.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 391.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 144.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 330.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 222.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 869.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 182.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 492.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.08e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 58.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 49.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + 548.T + 9.12e5T^{2} \) |
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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
−8.555514672550973385478446212687, −8.049836288445257562898490781114, −7.57483918977593649559800555850, −6.77189670865659367195735236127, −5.59458438935453127320308434118, −4.17804949608739138510811342801, −3.71697083805300250641618935661, −2.62015889000868437957593674107, −2.07739927873703168846521341017, −0.63601936744905086107375945217,
0.63601936744905086107375945217, 2.07739927873703168846521341017, 2.62015889000868437957593674107, 3.71697083805300250641618935661, 4.17804949608739138510811342801, 5.59458438935453127320308434118, 6.77189670865659367195735236127, 7.57483918977593649559800555850, 8.049836288445257562898490781114, 8.555514672550973385478446212687
