L(s) = 1 | + 1.72·2-s + 6.89·3-s − 5.01·4-s − 20.8·5-s + 11.9·6-s − 7.56·7-s − 22.4·8-s + 20.5·9-s − 35.9·10-s + 4.40·11-s − 34.5·12-s − 13.0·14-s − 143.·15-s + 1.27·16-s − 73.0·17-s + 35.5·18-s − 55.9·19-s + 104.·20-s − 52.1·21-s + 7.60·22-s − 33.6·23-s − 155.·24-s + 308.·25-s − 44.4·27-s + 37.9·28-s + 121.·29-s − 248.·30-s + ⋯ |
L(s) = 1 | + 0.610·2-s + 1.32·3-s − 0.626·4-s − 1.86·5-s + 0.810·6-s − 0.408·7-s − 0.993·8-s + 0.761·9-s − 1.13·10-s + 0.120·11-s − 0.831·12-s − 0.249·14-s − 2.47·15-s + 0.0199·16-s − 1.04·17-s + 0.464·18-s − 0.675·19-s + 1.16·20-s − 0.542·21-s + 0.0737·22-s − 0.304·23-s − 1.31·24-s + 2.47·25-s − 0.316·27-s + 0.256·28-s + 0.777·29-s − 1.51·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
good | 2 | \( 1 - 1.72T + 8T^{2} \) |
| 3 | \( 1 - 6.89T + 27T^{2} \) |
| 5 | \( 1 + 20.8T + 125T^{2} \) |
| 7 | \( 1 + 7.56T + 343T^{2} \) |
| 11 | \( 1 - 4.40T + 1.33e3T^{2} \) |
| 17 | \( 1 + 73.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 55.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 33.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 84.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 171.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 93.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 441.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 272.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 480.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 350.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 484.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 967.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 402.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 351.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 820.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 192.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 813.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 788.T + 9.12e5T^{2} \) |
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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
−12.16812455910547595100726160925, −11.00781766711374580678395097200, −9.444199515001512156907957427716, −8.533501220661045299739017416930, −7.996697905250206589264176345475, −6.64681847461118279845310507779, −4.60553623703116257751245642097, −3.83990001373111596933014957955, −2.93564508222258030457684441275, 0,
2.93564508222258030457684441275, 3.83990001373111596933014957955, 4.60553623703116257751245642097, 6.64681847461118279845310507779, 7.996697905250206589264176345475, 8.533501220661045299739017416930, 9.444199515001512156907957427716, 11.00781766711374580678395097200, 12.16812455910547595100726160925