| L(s) = 1 | + 0.819·2-s − 0.990·3-s − 1.32·4-s − 5-s − 0.811·6-s + 2.45·7-s − 2.72·8-s − 2.01·9-s − 0.819·10-s − 3.82·11-s + 1.31·12-s − 5.06·13-s + 2.00·14-s + 0.990·15-s + 0.418·16-s + 2.13·17-s − 1.65·18-s + 1.70·19-s + 1.32·20-s − 2.42·21-s − 3.13·22-s + 5.18·23-s + 2.70·24-s + 25-s − 4.15·26-s + 4.96·27-s − 3.25·28-s + ⋯ |
| L(s) = 1 | + 0.579·2-s − 0.571·3-s − 0.663·4-s − 0.447·5-s − 0.331·6-s + 0.926·7-s − 0.964·8-s − 0.673·9-s − 0.259·10-s − 1.15·11-s + 0.379·12-s − 1.40·13-s + 0.536·14-s + 0.255·15-s + 0.104·16-s + 0.517·17-s − 0.390·18-s + 0.392·19-s + 0.296·20-s − 0.529·21-s − 0.669·22-s + 1.08·23-s + 0.551·24-s + 0.200·25-s − 0.814·26-s + 0.956·27-s − 0.614·28-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
| \(L(1)\) |
\(\approx\) |
\(0.9681182498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9681182498\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.819T + 2T^{2} \) |
| 3 | \( 1 + 0.990T + 3T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 + 5.06T + 13T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 23 | \( 1 - 5.18T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + 7.68T + 31T^{2} \) |
| 37 | \( 1 - 9.67T + 37T^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 43 | \( 1 + 0.781T + 43T^{2} \) |
| 47 | \( 1 + 5.22T + 47T^{2} \) |
| 53 | \( 1 - 7.13T + 53T^{2} \) |
| 59 | \( 1 - 2.35T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 + 2.53T + 67T^{2} \) |
| 71 | \( 1 - 5.25T + 71T^{2} \) |
| 73 | \( 1 + 2.44T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 6.57T + 89T^{2} \) |
| 97 | \( 1 + 1.26T + 97T^{2} \) |
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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
−9.127059062336239505577344186219, −8.205695232511894366804518338679, −7.74404050041049579545457609352, −6.73450825002308958265504486180, −5.37346178353786284005635540056, −5.26791435193979091154939831481, −4.57891053934448607018395615243, −3.36547195868078671683317074632, −2.48237782466724224671710236019, −0.59944232565108132448248345127,
0.59944232565108132448248345127, 2.48237782466724224671710236019, 3.36547195868078671683317074632, 4.57891053934448607018395615243, 5.26791435193979091154939831481, 5.37346178353786284005635540056, 6.73450825002308958265504486180, 7.74404050041049579545457609352, 8.205695232511894366804518338679, 9.127059062336239505577344186219
