| L(s) = 1 | + 2.51·2-s + 3-s + 4.33·4-s + 2.41·5-s + 2.51·6-s + 5.87·8-s + 9-s + 6.08·10-s + 1.24·11-s + 4.33·12-s + 2.06·13-s + 2.41·15-s + 6.11·16-s − 3.31·17-s + 2.51·18-s − 3.61·19-s + 10.4·20-s + 3.14·22-s − 23-s + 5.87·24-s + 0.848·25-s + 5.20·26-s + 27-s + 4.27·29-s + 6.08·30-s − 7.97·31-s + 3.64·32-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 0.577·3-s + 2.16·4-s + 1.08·5-s + 1.02·6-s + 2.07·8-s + 0.333·9-s + 1.92·10-s + 0.376·11-s + 1.25·12-s + 0.573·13-s + 0.624·15-s + 1.52·16-s − 0.802·17-s + 0.593·18-s − 0.830·19-s + 2.34·20-s + 0.669·22-s − 0.208·23-s + 1.19·24-s + 0.169·25-s + 1.02·26-s + 0.192·27-s + 0.794·29-s + 1.11·30-s − 1.43·31-s + 0.644·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.539950693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.539950693\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 5 | \( 1 - 2.41T + 5T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 + 3.31T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 + 7.97T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 + 0.885T + 41T^{2} \) |
| 43 | \( 1 - 9.94T + 43T^{2} \) |
| 47 | \( 1 + 5.69T + 47T^{2} \) |
| 53 | \( 1 + 2.94T + 53T^{2} \) |
| 59 | \( 1 - 2.02T + 59T^{2} \) |
| 61 | \( 1 + 7.38T + 61T^{2} \) |
| 67 | \( 1 - 8.28T + 67T^{2} \) |
| 71 | \( 1 + 8.27T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−8.668102571648214167185575801071, −7.58997749103792210723494176751, −6.74223383840270181274705677135, −6.14987278207963455186048313596, −5.64819484013356313581872618854, −4.59988617769549932785918583162, −4.07471408342601208663171015534, −3.13635093418087975154235134300, −2.29992415055520986592992191184, −1.63646277832710497953569223805,
1.63646277832710497953569223805, 2.29992415055520986592992191184, 3.13635093418087975154235134300, 4.07471408342601208663171015534, 4.59988617769549932785918583162, 5.64819484013356313581872618854, 6.14987278207963455186048313596, 6.74223383840270181274705677135, 7.58997749103792210723494176751, 8.668102571648214167185575801071
