| L(s) = 1 | − 2-s + 4-s − 1.32·5-s − 0.0838·7-s − 8-s + 1.32·10-s − 11-s − 1.48·13-s + 0.0838·14-s + 16-s − 6.36·17-s − 4.66·19-s − 1.32·20-s + 22-s − 7.20·23-s − 3.24·25-s + 1.48·26-s − 0.0838·28-s − 5.19·29-s + 2·31-s − 32-s + 6.36·34-s + 0.111·35-s − 6.59·37-s + 4.66·38-s + 1.32·40-s − 0.194·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.592·5-s − 0.0317·7-s − 0.353·8-s + 0.418·10-s − 0.301·11-s − 0.412·13-s + 0.0224·14-s + 0.250·16-s − 1.54·17-s − 1.07·19-s − 0.296·20-s + 0.213·22-s − 1.50·23-s − 0.649·25-s + 0.291·26-s − 0.0158·28-s − 0.965·29-s + 0.359·31-s − 0.176·32-s + 1.09·34-s + 0.0187·35-s − 1.08·37-s + 0.757·38-s + 0.209·40-s − 0.0304·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2718528777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2718528777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 5 | \( 1 + 1.32T + 5T^{2} \) |
| 7 | \( 1 + 0.0838T + 7T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 + 6.36T + 17T^{2} \) |
| 19 | \( 1 + 4.66T + 19T^{2} \) |
| 23 | \( 1 + 7.20T + 23T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 6.59T + 37T^{2} \) |
| 41 | \( 1 + 0.194T + 41T^{2} \) |
| 43 | \( 1 - 5.77T + 43T^{2} \) |
| 53 | \( 1 - 2.19T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 6.47T + 61T^{2} \) |
| 67 | \( 1 - 7.28T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 2.03T + 73T^{2} \) |
| 79 | \( 1 + 3.93T + 79T^{2} \) |
| 83 | \( 1 + 2.95T + 83T^{2} \) |
| 89 | \( 1 - 4.33T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.66957082403342131998152142795, −7.27859324624220463730838854037, −6.37324717273900207053526506686, −5.94020866808281275108881988416, −4.82624814053493731706706517215, −4.19208692068721202011517569059, −3.45024237299633284750977923757, −2.30666746279693199264882450677, −1.86934566299586465691217238322, −0.26043366574268795641313926252,
0.26043366574268795641313926252, 1.86934566299586465691217238322, 2.30666746279693199264882450677, 3.45024237299633284750977923757, 4.19208692068721202011517569059, 4.82624814053493731706706517215, 5.94020866808281275108881988416, 6.37324717273900207053526506686, 7.27859324624220463730838854037, 7.66957082403342131998152142795
