| L(s) = 1 | − 2-s − 2.04·3-s + 4-s + 0.868·5-s + 2.04·6-s − 8-s + 1.17·9-s − 0.868·10-s − 4.52·11-s − 2.04·12-s − 4.81·13-s − 1.77·15-s + 16-s + 0.259·17-s − 1.17·18-s − 1.04·19-s + 0.868·20-s + 4.52·22-s − 7.24·23-s + 2.04·24-s − 4.24·25-s + 4.81·26-s + 3.73·27-s + 29-s + 1.77·30-s + 2.40·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.17·3-s + 0.5·4-s + 0.388·5-s + 0.833·6-s − 0.353·8-s + 0.390·9-s − 0.274·10-s − 1.36·11-s − 0.589·12-s − 1.33·13-s − 0.457·15-s + 0.250·16-s + 0.0630·17-s − 0.276·18-s − 0.239·19-s + 0.194·20-s + 0.964·22-s − 1.51·23-s + 0.416·24-s − 0.849·25-s + 0.943·26-s + 0.718·27-s + 0.185·29-s + 0.323·30-s + 0.432·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3393523532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3393523532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.04T + 3T^{2} \) |
| 5 | \( 1 - 0.868T + 5T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 - 0.259T + 17T^{2} \) |
| 19 | \( 1 + 1.04T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 31 | \( 1 - 2.40T + 31T^{2} \) |
| 37 | \( 1 + 7.46T + 37T^{2} \) |
| 41 | \( 1 + 5.52T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 5.23T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 6.20T + 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 6.90T + 83T^{2} \) |
| 89 | \( 1 - 1.73T + 89T^{2} \) |
| 97 | \( 1 + 5.00T + 97T^{2} \) |
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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
−8.789012239434773502534064081367, −7.936938626085738569805485934258, −7.33838844992712995699607959084, −6.48665170168451239229270015672, −5.63897013278456296102711063898, −5.28361032796332572303330852347, −4.23247650291649566822786359208, −2.74759844545734438500248455385, −1.99326076278148074982073417573, −0.39815199165891078136141415657,
0.39815199165891078136141415657, 1.99326076278148074982073417573, 2.74759844545734438500248455385, 4.23247650291649566822786359208, 5.28361032796332572303330852347, 5.63897013278456296102711063898, 6.48665170168451239229270015672, 7.33838844992712995699607959084, 7.936938626085738569805485934258, 8.789012239434773502534064081367
