| L(s) = 1 | − 0.237·2-s − 3-s − 1.94·4-s + 0.237·6-s − 1.64·7-s + 0.935·8-s + 9-s + 5.84·11-s + 1.94·12-s + 4.61·13-s + 0.390·14-s + 3.66·16-s + 5.67·17-s − 0.237·18-s − 6.71·19-s + 1.64·21-s − 1.38·22-s + 6.07·23-s − 0.935·24-s − 1.09·26-s − 27-s + 3.20·28-s + 3.23·29-s − 6.81·31-s − 2.74·32-s − 5.84·33-s − 1.34·34-s + ⋯ |
| L(s) = 1 | − 0.167·2-s − 0.577·3-s − 0.971·4-s + 0.0968·6-s − 0.622·7-s + 0.330·8-s + 0.333·9-s + 1.76·11-s + 0.561·12-s + 1.27·13-s + 0.104·14-s + 0.916·16-s + 1.37·17-s − 0.0559·18-s − 1.54·19-s + 0.359·21-s − 0.295·22-s + 1.26·23-s − 0.190·24-s − 0.214·26-s − 0.192·27-s + 0.604·28-s + 0.600·29-s − 1.22·31-s − 0.484·32-s − 1.01·33-s − 0.230·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.199410484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.199410484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 0.237T + 2T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 - 5.84T + 11T^{2} \) |
| 13 | \( 1 - 4.61T + 13T^{2} \) |
| 17 | \( 1 - 5.67T + 17T^{2} \) |
| 19 | \( 1 + 6.71T + 19T^{2} \) |
| 23 | \( 1 - 6.07T + 23T^{2} \) |
| 29 | \( 1 - 3.23T + 29T^{2} \) |
| 31 | \( 1 + 6.81T + 31T^{2} \) |
| 37 | \( 1 + 5.84T + 37T^{2} \) |
| 41 | \( 1 + 2.60T + 41T^{2} \) |
| 43 | \( 1 + 0.899T + 43T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 - 6.93T + 59T^{2} \) |
| 61 | \( 1 + 6.54T + 61T^{2} \) |
| 67 | \( 1 - 9.16T + 67T^{2} \) |
| 71 | \( 1 - 3.98T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 7.03T + 89T^{2} \) |
| 97 | \( 1 + 8.59T + 97T^{2} \) |
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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
−8.730384685123854153448636822809, −8.012986812495194300646577243194, −6.80891908289831514000523886241, −6.43128559024784776152745787218, −5.58475608559302111917758045707, −4.77327371634645298372999038530, −3.71185550161639886041529030545, −3.54764114881775072239638008476, −1.58798079713987883517378571123, −0.75194083378277500445155014235,
0.75194083378277500445155014235, 1.58798079713987883517378571123, 3.54764114881775072239638008476, 3.71185550161639886041529030545, 4.77327371634645298372999038530, 5.58475608559302111917758045707, 6.43128559024784776152745787218, 6.80891908289831514000523886241, 8.012986812495194300646577243194, 8.730384685123854153448636822809
