L(s) = 1 | + 1.85·2-s − 2.98·3-s + 1.44·4-s − 3.13·5-s − 5.54·6-s + 3.56·7-s − 1.03·8-s + 5.91·9-s − 5.81·10-s − 11-s − 4.30·12-s + 3.92·13-s + 6.61·14-s + 9.36·15-s − 4.80·16-s + 17-s + 10.9·18-s − 2.74·19-s − 4.52·20-s − 10.6·21-s − 1.85·22-s + 4.40·23-s + 3.09·24-s + 4.83·25-s + 7.27·26-s − 8.71·27-s + 5.14·28-s + ⋯ |
L(s) = 1 | + 1.31·2-s − 1.72·3-s + 0.720·4-s − 1.40·5-s − 2.26·6-s + 1.34·7-s − 0.366·8-s + 1.97·9-s − 1.83·10-s − 0.301·11-s − 1.24·12-s + 1.08·13-s + 1.76·14-s + 2.41·15-s − 1.20·16-s + 0.242·17-s + 2.58·18-s − 0.629·19-s − 1.01·20-s − 2.32·21-s − 0.395·22-s + 0.918·23-s + 0.631·24-s + 0.967·25-s + 1.42·26-s − 1.67·27-s + 0.971·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.555405481\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.555405481\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 - 1.85T + 2T^{2} \) |
| 3 | \( 1 + 2.98T + 3T^{2} \) |
| 5 | \( 1 + 3.13T + 5T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 13 | \( 1 - 3.92T + 13T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 23 | \( 1 - 4.40T + 23T^{2} \) |
| 29 | \( 1 - 6.77T + 29T^{2} \) |
| 31 | \( 1 + 1.01T + 31T^{2} \) |
| 37 | \( 1 + 2.54T + 37T^{2} \) |
| 41 | \( 1 + 8.54T + 41T^{2} \) |
| 47 | \( 1 - 3.28T + 47T^{2} \) |
| 53 | \( 1 - 1.03T + 53T^{2} \) |
| 59 | \( 1 + 3.45T + 59T^{2} \) |
| 61 | \( 1 + 7.27T + 61T^{2} \) |
| 67 | \( 1 + 1.00T + 67T^{2} \) |
| 71 | \( 1 + 3.94T + 71T^{2} \) |
| 73 | \( 1 + 5.32T + 73T^{2} \) |
| 79 | \( 1 - 2.93T + 79T^{2} \) |
| 83 | \( 1 - 2.66T + 83T^{2} \) |
| 89 | \( 1 - 5.04T + 89T^{2} \) |
| 97 | \( 1 + 1.72T + 97T^{2} \) |
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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
−7.58949098434812124553892346290, −6.84600113212062027128241853566, −6.26254790500106155145893060994, −5.51001546518352293196076382756, −4.89573574499239808694435634091, −4.54853248250081442778993615762, −3.92053410267138129507811663587, −3.09936606657681409691366353524, −1.61218393804231996576314500687, −0.57583563791570222381108471669,
0.57583563791570222381108471669, 1.61218393804231996576314500687, 3.09936606657681409691366353524, 3.92053410267138129507811663587, 4.54853248250081442778993615762, 4.89573574499239808694435634091, 5.51001546518352293196076382756, 6.26254790500106155145893060994, 6.84600113212062027128241853566, 7.58949098434812124553892346290