| L(s) = 1 | − 4.70·2-s − 3·3-s + 14.1·4-s + 14.1·6-s + 16.2·7-s − 28.7·8-s + 9·9-s − 40.2·11-s − 42.3·12-s − 19.7·13-s − 76.2·14-s + 22.1·16-s − 83.0·17-s − 42.3·18-s − 48.8·19-s − 48.6·21-s + 189.·22-s − 1.61·23-s + 86.1·24-s + 93.0·26-s − 27·27-s + 228.·28-s − 24.5·29-s − 12.4·31-s + 125.·32-s + 120.·33-s + 390.·34-s + ⋯ |
| L(s) = 1 | − 1.66·2-s − 0.577·3-s + 1.76·4-s + 0.959·6-s + 0.875·7-s − 1.26·8-s + 0.333·9-s − 1.10·11-s − 1.01·12-s − 0.422·13-s − 1.45·14-s + 0.345·16-s − 1.18·17-s − 0.554·18-s − 0.589·19-s − 0.505·21-s + 1.83·22-s − 0.0146·23-s + 0.732·24-s + 0.701·26-s − 0.192·27-s + 1.54·28-s − 0.157·29-s − 0.0719·31-s + 0.694·32-s + 0.636·33-s + 1.96·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 4.70T + 8T^{2} \) |
| 7 | \( 1 - 16.2T + 343T^{2} \) |
| 11 | \( 1 + 40.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 83.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.61T + 1.21e4T^{2} \) |
| 29 | \( 1 + 24.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 325.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 242.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 204.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 61.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 112.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 477.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 558.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 558.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 96.9T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−13.36192549531229355700919889602, −11.88584040810664907560401021818, −10.89247678156219772689310476860, −10.24068989855256366584006862879, −8.839995518972022660707056741285, −7.88239580595118474581187244409, −6.75418587007553542634970726691, −4.97440237028538526967341119387, −2.00098149295913015009506559826, 0,
2.00098149295913015009506559826, 4.97440237028538526967341119387, 6.75418587007553542634970726691, 7.88239580595118474581187244409, 8.839995518972022660707056741285, 10.24068989855256366584006862879, 10.89247678156219772689310476860, 11.88584040810664907560401021818, 13.36192549531229355700919889602
