| L(s) = 1 | − 1.06·2-s − 1.17·3-s − 0.872·4-s + 1.24·6-s + 7-s + 3.05·8-s − 1.62·9-s − 11-s + 1.02·12-s − 6.82·13-s − 1.06·14-s − 1.49·16-s + 2.26·17-s + 1.73·18-s + 6.41·19-s − 1.17·21-s + 1.06·22-s − 9.35·23-s − 3.57·24-s + 7.25·26-s + 5.41·27-s − 0.872·28-s + 4.96·29-s − 0.856·31-s − 4.51·32-s + 1.17·33-s − 2.40·34-s + ⋯ |
| L(s) = 1 | − 0.750·2-s − 0.675·3-s − 0.436·4-s + 0.507·6-s + 0.377·7-s + 1.07·8-s − 0.543·9-s − 0.301·11-s + 0.294·12-s − 1.89·13-s − 0.283·14-s − 0.373·16-s + 0.548·17-s + 0.408·18-s + 1.47·19-s − 0.255·21-s + 0.226·22-s − 1.95·23-s − 0.728·24-s + 1.42·26-s + 1.04·27-s − 0.164·28-s + 0.921·29-s − 0.153·31-s − 0.797·32-s + 0.203·33-s − 0.412·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4575837440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4575837440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.06T + 2T^{2} \) |
| 3 | \( 1 + 1.17T + 3T^{2} \) |
| 13 | \( 1 + 6.82T + 13T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 23 | \( 1 + 9.35T + 23T^{2} \) |
| 29 | \( 1 - 4.96T + 29T^{2} \) |
| 31 | \( 1 + 0.856T + 31T^{2} \) |
| 37 | \( 1 + 6.61T + 37T^{2} \) |
| 41 | \( 1 - 0.527T + 41T^{2} \) |
| 43 | \( 1 + 4.95T + 43T^{2} \) |
| 47 | \( 1 + 1.24T + 47T^{2} \) |
| 53 | \( 1 - 3.97T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 3.40T + 61T^{2} \) |
| 67 | \( 1 + 5.32T + 67T^{2} \) |
| 71 | \( 1 - 4.43T + 71T^{2} \) |
| 73 | \( 1 + 1.06T + 73T^{2} \) |
| 79 | \( 1 + 0.966T + 79T^{2} \) |
| 83 | \( 1 - 6.07T + 83T^{2} \) |
| 89 | \( 1 + 3.55T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−9.301771734868003958368580339180, −8.354463967360995451756024067519, −7.74708891509894180281428521142, −7.12222031425597023963979550940, −5.86511631501615276350032502406, −5.12675725999765132100403220278, −4.60849071714499378158244555484, −3.23139689827031753520269960749, −1.95648856448256419557939359562, −0.51744779433425574837941278036,
0.51744779433425574837941278036, 1.95648856448256419557939359562, 3.23139689827031753520269960749, 4.60849071714499378158244555484, 5.12675725999765132100403220278, 5.86511631501615276350032502406, 7.12222031425597023963979550940, 7.74708891509894180281428521142, 8.354463967360995451756024067519, 9.301771734868003958368580339180
