| L(s) = 1 | + 2.53·2-s − 6.96·3-s − 1.57·4-s + 11.2·5-s − 17.6·6-s + 27.5·7-s − 24.2·8-s + 21.5·9-s + 28.4·10-s + 55.7·11-s + 11.0·12-s + 72.2·13-s + 69.6·14-s − 78.2·15-s − 48.8·16-s + 74.4·17-s + 54.5·18-s + 99.5·19-s − 17.7·20-s − 191.·21-s + 141.·22-s − 33.8·23-s + 169.·24-s + 1.00·25-s + 183.·26-s + 38.0·27-s − 43.4·28-s + ⋯ |
| L(s) = 1 | + 0.895·2-s − 1.34·3-s − 0.197·4-s + 1.00·5-s − 1.20·6-s + 1.48·7-s − 1.07·8-s + 0.797·9-s + 0.899·10-s + 1.52·11-s + 0.264·12-s + 1.54·13-s + 1.33·14-s − 1.34·15-s − 0.763·16-s + 1.06·17-s + 0.714·18-s + 1.20·19-s − 0.198·20-s − 1.99·21-s + 1.36·22-s − 0.307·23-s + 1.43·24-s + 0.00800·25-s + 1.38·26-s + 0.271·27-s − 0.293·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.772979013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.772979013\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 2.53T + 8T^{2} \) |
| 3 | \( 1 + 6.96T + 27T^{2} \) |
| 5 | \( 1 - 11.2T + 125T^{2} \) |
| 7 | \( 1 - 27.5T + 343T^{2} \) |
| 11 | \( 1 - 55.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 99.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 33.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 219.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 155.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 125.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 235.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 361.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 267.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 14.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 635.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 719.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 238.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 253.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 263.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 219.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 26.7T + 9.12e5T^{2} \) |
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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
−8.873376784693418763537172319560, −8.204107593893869481950305654757, −6.83430445660944177448821828513, −6.00904781902663033028069950607, −5.72874064972787749813760391432, −4.93713875317466020656135165558, −4.25467211284052399057064072097, −3.22570621824923112189722066588, −1.46822302374666872906608829922, −1.03038107327210238484566232438,
1.03038107327210238484566232438, 1.46822302374666872906608829922, 3.22570621824923112189722066588, 4.25467211284052399057064072097, 4.93713875317466020656135165558, 5.72874064972787749813760391432, 6.00904781902663033028069950607, 6.83430445660944177448821828513, 8.204107593893869481950305654757, 8.873376784693418763537172319560
