| L(s) = 1 | − 0.686·2-s + 2.32·3-s − 1.52·4-s + 1.52·5-s − 1.59·6-s + 3.39·7-s + 2.42·8-s + 2.40·9-s − 1.04·10-s + 2.77·11-s − 3.55·12-s − 0.00660·13-s − 2.33·14-s + 3.55·15-s + 1.39·16-s + 3.74·17-s − 1.64·18-s − 4.86·19-s − 2.33·20-s + 7.89·21-s − 1.90·22-s + 8.82·23-s + 5.62·24-s − 2.66·25-s + 0.00452·26-s − 1.39·27-s − 5.19·28-s + ⋯ |
| L(s) = 1 | − 0.485·2-s + 1.34·3-s − 0.764·4-s + 0.683·5-s − 0.650·6-s + 1.28·7-s + 0.856·8-s + 0.800·9-s − 0.331·10-s + 0.837·11-s − 1.02·12-s − 0.00183·13-s − 0.622·14-s + 0.917·15-s + 0.349·16-s + 0.907·17-s − 0.388·18-s − 1.11·19-s − 0.522·20-s + 1.72·21-s − 0.406·22-s + 1.83·23-s + 1.14·24-s − 0.532·25-s + 0.000888·26-s − 0.267·27-s − 0.981·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.766612961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.766612961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2671 | \( 1 - T \) |
| good | 2 | \( 1 + 0.686T + 2T^{2} \) |
| 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 - 1.52T + 5T^{2} \) |
| 7 | \( 1 - 3.39T + 7T^{2} \) |
| 11 | \( 1 - 2.77T + 11T^{2} \) |
| 13 | \( 1 + 0.00660T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 + 4.86T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 + 6.09T + 29T^{2} \) |
| 31 | \( 1 - 7.98T + 31T^{2} \) |
| 37 | \( 1 - 8.78T + 37T^{2} \) |
| 41 | \( 1 - 1.02T + 41T^{2} \) |
| 43 | \( 1 + 2.07T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 5.58T + 53T^{2} \) |
| 59 | \( 1 + 1.30T + 59T^{2} \) |
| 61 | \( 1 - 2.28T + 61T^{2} \) |
| 67 | \( 1 + 2.93T + 67T^{2} \) |
| 71 | \( 1 - 5.01T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 5.10T + 79T^{2} \) |
| 83 | \( 1 + 9.18T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−8.791082794540746513367053405390, −8.180805021407231093289724580105, −7.84245326916238622157552168396, −6.79082724523559942744519609399, −5.65279758634432685008700649294, −4.75638224618742020800953358131, −4.07715863484224817363852281256, −3.05707613620474110420818473668, −1.91705780365486164171445638887, −1.21515960487672589118748885433,
1.21515960487672589118748885433, 1.91705780365486164171445638887, 3.05707613620474110420818473668, 4.07715863484224817363852281256, 4.75638224618742020800953358131, 5.65279758634432685008700649294, 6.79082724523559942744519609399, 7.84245326916238622157552168396, 8.180805021407231093289724580105, 8.791082794540746513367053405390
