| L(s) = 1 | − 1.63·2-s + 0.669·4-s − 0.505·7-s + 2.17·8-s + 3.10·11-s + 6.23·13-s + 0.825·14-s − 4.89·16-s + 6.10·17-s − 5.57·19-s − 5.06·22-s + 3.82·23-s − 10.1·26-s − 0.338·28-s + 2.45·29-s + 4.22·31-s + 3.64·32-s − 9.96·34-s − 6.72·37-s + 9.10·38-s − 5.44·41-s − 1.32·43-s + 2.07·44-s − 6.24·46-s + 3.70·47-s − 6.74·49-s + 4.17·52-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.334·4-s − 0.191·7-s + 0.768·8-s + 0.934·11-s + 1.73·13-s + 0.220·14-s − 1.22·16-s + 1.47·17-s − 1.27·19-s − 1.07·22-s + 0.797·23-s − 1.99·26-s − 0.0638·28-s + 0.456·29-s + 0.759·31-s + 0.643·32-s − 1.70·34-s − 1.10·37-s + 1.47·38-s − 0.849·41-s − 0.202·43-s + 0.312·44-s − 0.921·46-s + 0.540·47-s − 0.963·49-s + 0.578·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.046899625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.046899625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.63T + 2T^{2} \) |
| 7 | \( 1 + 0.505T + 7T^{2} \) |
| 11 | \( 1 - 3.10T + 11T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 - 6.10T + 17T^{2} \) |
| 19 | \( 1 + 5.57T + 19T^{2} \) |
| 23 | \( 1 - 3.82T + 23T^{2} \) |
| 29 | \( 1 - 2.45T + 29T^{2} \) |
| 31 | \( 1 - 4.22T + 31T^{2} \) |
| 37 | \( 1 + 6.72T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 - 3.70T + 47T^{2} \) |
| 53 | \( 1 + 2.54T + 53T^{2} \) |
| 59 | \( 1 - 2.88T + 59T^{2} \) |
| 61 | \( 1 + 2.84T + 61T^{2} \) |
| 67 | \( 1 + 2.40T + 67T^{2} \) |
| 71 | \( 1 - 5.54T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 3.40T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 3.38T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−9.041997439480322655012827793477, −8.464982787180121313729999809910, −7.948388604303912798388723290402, −6.78876079769744541553254763541, −6.33894003377905974872338788770, −5.17552455575905478013320325446, −4.08327006510256462781802320264, −3.30360244554730699438722768821, −1.70440772349425881315142198940, −0.886399801437427913467559339949,
0.886399801437427913467559339949, 1.70440772349425881315142198940, 3.30360244554730699438722768821, 4.08327006510256462781802320264, 5.17552455575905478013320325446, 6.33894003377905974872338788770, 6.78876079769744541553254763541, 7.948388604303912798388723290402, 8.464982787180121313729999809910, 9.041997439480322655012827793477
