| L(s) = 1 | − 2-s + 2.18·3-s + 4-s − 3.74·5-s − 2.18·6-s − 3.22·7-s − 8-s + 1.78·9-s + 3.74·10-s − 5.97·11-s + 2.18·12-s − 6.04·13-s + 3.22·14-s − 8.19·15-s + 16-s − 6.46·17-s − 1.78·18-s + 4.65·19-s − 3.74·20-s − 7.06·21-s + 5.97·22-s + 7.81·23-s − 2.18·24-s + 9.01·25-s + 6.04·26-s − 2.64·27-s − 3.22·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.26·3-s + 0.5·4-s − 1.67·5-s − 0.893·6-s − 1.21·7-s − 0.353·8-s + 0.596·9-s + 1.18·10-s − 1.80·11-s + 0.631·12-s − 1.67·13-s + 0.862·14-s − 2.11·15-s + 0.250·16-s − 1.56·17-s − 0.421·18-s + 1.06·19-s − 0.836·20-s − 1.54·21-s + 1.27·22-s + 1.63·23-s − 0.446·24-s + 1.80·25-s + 1.18·26-s − 0.509·27-s − 0.609·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2870612484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2870612484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 + 3.74T + 5T^{2} \) |
| 7 | \( 1 + 3.22T + 7T^{2} \) |
| 11 | \( 1 + 5.97T + 11T^{2} \) |
| 13 | \( 1 + 6.04T + 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 - 4.65T + 19T^{2} \) |
| 23 | \( 1 - 7.81T + 23T^{2} \) |
| 29 | \( 1 + 7.38T + 29T^{2} \) |
| 31 | \( 1 - 6.58T + 31T^{2} \) |
| 37 | \( 1 + 1.29T + 37T^{2} \) |
| 41 | \( 1 - 1.11T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 5.54T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 1.71T + 59T^{2} \) |
| 61 | \( 1 - 7.61T + 61T^{2} \) |
| 67 | \( 1 - 5.36T + 67T^{2} \) |
| 71 | \( 1 + 6.49T + 71T^{2} \) |
| 73 | \( 1 + 6.69T + 73T^{2} \) |
| 79 | \( 1 + 4.90T + 79T^{2} \) |
| 83 | \( 1 + 8.90T + 83T^{2} \) |
| 89 | \( 1 - 2.14T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.475117948921666466586173920580, −7.71260254497072960246552953567, −7.27891380397656031134482954323, −6.85463656318492193773885871108, −5.31625468706607634489793744743, −4.54376570752381102605631108635, −3.38804502176949557777743036846, −2.93981739943610950213262625046, −2.35504597242430005774977557577, −0.28967697131955603649600785733,
0.28967697131955603649600785733, 2.35504597242430005774977557577, 2.93981739943610950213262625046, 3.38804502176949557777743036846, 4.54376570752381102605631108635, 5.31625468706607634489793744743, 6.85463656318492193773885871108, 7.27891380397656031134482954323, 7.71260254497072960246552953567, 8.475117948921666466586173920580
