| L(s) = 1 | + 4.08·2-s + 3·3-s + 8.69·4-s − 16.7·5-s + 12.2·6-s − 27.0·7-s + 2.81·8-s + 9·9-s − 68.2·10-s − 49.7·11-s + 26.0·12-s + 48.6·13-s − 110.·14-s − 50.1·15-s − 58.0·16-s + 19.8·17-s + 36.7·18-s + 12.7·19-s − 145.·20-s − 81.0·21-s − 203.·22-s + 23·23-s + 8.45·24-s + 153.·25-s + 198.·26-s + 27·27-s − 234.·28-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 0.577·3-s + 1.08·4-s − 1.49·5-s + 0.833·6-s − 1.45·7-s + 0.124·8-s + 0.333·9-s − 2.15·10-s − 1.36·11-s + 0.627·12-s + 1.03·13-s − 2.10·14-s − 0.862·15-s − 0.906·16-s + 0.282·17-s + 0.481·18-s + 0.153·19-s − 1.62·20-s − 0.842·21-s − 1.96·22-s + 0.208·23-s + 0.0719·24-s + 1.23·25-s + 1.50·26-s + 0.192·27-s − 1.58·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.561075015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.561075015\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 4.08T + 8T^{2} \) |
| 5 | \( 1 + 16.7T + 125T^{2} \) |
| 7 | \( 1 + 27.0T + 343T^{2} \) |
| 11 | \( 1 + 49.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 12.7T + 6.85e3T^{2} \) |
| 31 | \( 1 - 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 108.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 260.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 447.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 201.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 59.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 353.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 13.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 700.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 344.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 653.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 93.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 586.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 491.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
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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
−8.591451927762650647715342395532, −7.932606848287244234212839330034, −7.09135703865802521365592199210, −6.35991429123137436904555605264, −5.47553407653993661139489828836, −4.50277076186368065325404185420, −3.71020067460459918797505125807, −3.25575031574120725468821565468, −2.57041846254190274486774582710, −0.54265657002405803035694182929,
0.54265657002405803035694182929, 2.57041846254190274486774582710, 3.25575031574120725468821565468, 3.71020067460459918797505125807, 4.50277076186368065325404185420, 5.47553407653993661139489828836, 6.35991429123137436904555605264, 7.09135703865802521365592199210, 7.932606848287244234212839330034, 8.591451927762650647715342395532
