| L(s) = 1 | + 2.17·2-s + 0.884·3-s − 3.25·4-s − 14.1·5-s + 1.92·6-s + 17.3·7-s − 24.5·8-s − 26.2·9-s − 30.8·10-s − 72.6·11-s − 2.87·12-s + 0.996·13-s + 37.8·14-s − 12.5·15-s − 27.3·16-s + 98.1·17-s − 57.1·18-s − 31.6·19-s + 46.1·20-s + 15.3·21-s − 158.·22-s + 45.8·23-s − 21.6·24-s + 75.9·25-s + 2.17·26-s − 47.0·27-s − 56.4·28-s + ⋯ |
| L(s) = 1 | + 0.770·2-s + 0.170·3-s − 0.406·4-s − 1.26·5-s + 0.131·6-s + 0.937·7-s − 1.08·8-s − 0.971·9-s − 0.976·10-s − 1.99·11-s − 0.0691·12-s + 0.0212·13-s + 0.722·14-s − 0.215·15-s − 0.428·16-s + 1.40·17-s − 0.747·18-s − 0.381·19-s + 0.515·20-s + 0.159·21-s − 1.53·22-s + 0.416·23-s − 0.184·24-s + 0.607·25-s + 0.0163·26-s − 0.335·27-s − 0.381·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.291663748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.291663748\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 - 2.17T + 8T^{2} \) |
| 3 | \( 1 - 0.884T + 27T^{2} \) |
| 5 | \( 1 + 14.1T + 125T^{2} \) |
| 7 | \( 1 - 17.3T + 343T^{2} \) |
| 11 | \( 1 + 72.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.996T + 2.19e3T^{2} \) |
| 17 | \( 1 - 98.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 45.8T + 1.21e4T^{2} \) |
| 31 | \( 1 - 209.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 137.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 284.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 195.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 53.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 334.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 654.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 626.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 334.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 647.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 365.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 629.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 683.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 83.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + 16.4T + 9.12e5T^{2} \) |
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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
−9.882010199735325903980532737523, −8.595564904687120814334426909195, −8.052185758315292296232082802261, −7.62413802219111627420357393966, −5.95713013436767044677357269680, −5.14926964979454552784745350816, −4.54557639639310985075477799308, −3.38979037637614439792193503166, −2.66514376077812163568450691493, −0.53543475220889649989856973319,
0.53543475220889649989856973319, 2.66514376077812163568450691493, 3.38979037637614439792193503166, 4.54557639639310985075477799308, 5.14926964979454552784745350816, 5.95713013436767044677357269680, 7.62413802219111627420357393966, 8.052185758315292296232082802261, 8.595564904687120814334426909195, 9.882010199735325903980532737523
