L(s) = 1 | + 0.0686·2-s − 2.29·3-s − 1.99·4-s + 5-s − 0.157·6-s − 2.03·7-s − 0.274·8-s + 2.25·9-s + 0.0686·10-s + 1.26·11-s + 4.57·12-s − 4.81·13-s − 0.139·14-s − 2.29·15-s + 3.97·16-s − 2.38·17-s + 0.155·18-s − 7.50·19-s − 1.99·20-s + 4.66·21-s + 0.0868·22-s + 4.71·23-s + 0.629·24-s + 25-s − 0.330·26-s + 1.69·27-s + 4.05·28-s + ⋯ |
L(s) = 1 | + 0.0485·2-s − 1.32·3-s − 0.997·4-s + 0.447·5-s − 0.0643·6-s − 0.768·7-s − 0.0970·8-s + 0.753·9-s + 0.0217·10-s + 0.381·11-s + 1.32·12-s − 1.33·13-s − 0.0373·14-s − 0.592·15-s + 0.992·16-s − 0.579·17-s + 0.0365·18-s − 1.72·19-s − 0.446·20-s + 1.01·21-s + 0.0185·22-s + 0.982·23-s + 0.128·24-s + 0.200·25-s − 0.0648·26-s + 0.326·27-s + 0.766·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3234852381\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3234852381\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 - 0.0686T + 2T^{2} \) |
| 3 | \( 1 + 2.29T + 3T^{2} \) |
| 7 | \( 1 + 2.03T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 + 7.50T + 19T^{2} \) |
| 23 | \( 1 - 4.71T + 23T^{2} \) |
| 29 | \( 1 + 7.82T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 1.21T + 37T^{2} \) |
| 41 | \( 1 + 5.30T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 1.14T + 47T^{2} \) |
| 53 | \( 1 + 4.81T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 6.36T + 67T^{2} \) |
| 71 | \( 1 - 7.57T + 71T^{2} \) |
| 73 | \( 1 - 8.79T + 73T^{2} \) |
| 79 | \( 1 - 7.66T + 79T^{2} \) |
| 83 | \( 1 + 0.963T + 83T^{2} \) |
| 89 | \( 1 + 4.66T + 89T^{2} \) |
| 97 | \( 1 + 3.28T + 97T^{2} \) |
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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
−9.348972296595800083578481860755, −8.618361310493119573587647218120, −7.34484251587474060420281368487, −6.62160632437968034978958576906, −5.88312077847324028505218592056, −5.16842478023244391674257170364, −4.51794240362684592799918888713, −3.52180469135496158416484520292, −2.07379607098101125571244056541, −0.38613819092190072103578576928,
0.38613819092190072103578576928, 2.07379607098101125571244056541, 3.52180469135496158416484520292, 4.51794240362684592799918888713, 5.16842478023244391674257170364, 5.88312077847324028505218592056, 6.62160632437968034978958576906, 7.34484251587474060420281368487, 8.618361310493119573587647218120, 9.348972296595800083578481860755