| L(s) = 1 | − 2.66·3-s − 4.12·5-s + 4.21·7-s + 4.12·9-s + 11-s − 2.21·13-s + 11.0·15-s − 3.45·17-s + 19-s − 11.2·21-s + 5.45·23-s + 12.0·25-s − 3.00·27-s + 5.57·29-s − 7.00·31-s − 2.66·33-s − 17.3·35-s − 2.90·37-s + 5.90·39-s − 11.9·41-s − 1.46·43-s − 17.0·45-s − 7.58·47-s + 10.7·49-s + 9.22·51-s − 13.2·53-s − 4.12·55-s + ⋯ |
| L(s) = 1 | − 1.54·3-s − 1.84·5-s + 1.59·7-s + 1.37·9-s + 0.301·11-s − 0.614·13-s + 2.84·15-s − 0.837·17-s + 0.229·19-s − 2.45·21-s + 1.13·23-s + 2.40·25-s − 0.577·27-s + 1.03·29-s − 1.25·31-s − 0.464·33-s − 2.93·35-s − 0.478·37-s + 0.946·39-s − 1.86·41-s − 0.222·43-s − 2.53·45-s − 1.10·47-s + 1.53·49-s + 1.29·51-s − 1.81·53-s − 0.556·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6119879550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6119879550\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.66T + 3T^{2} \) |
| 5 | \( 1 + 4.12T + 5T^{2} \) |
| 7 | \( 1 - 4.21T + 7T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 + 3.45T + 17T^{2} \) |
| 23 | \( 1 - 5.45T + 23T^{2} \) |
| 29 | \( 1 - 5.57T + 29T^{2} \) |
| 31 | \( 1 + 7.00T + 31T^{2} \) |
| 37 | \( 1 + 2.90T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 + 4.79T + 59T^{2} \) |
| 61 | \( 1 - 8.90T + 61T^{2} \) |
| 67 | \( 1 + 1.30T + 67T^{2} \) |
| 71 | \( 1 - 6.80T + 71T^{2} \) |
| 73 | \( 1 + 1.45T + 73T^{2} \) |
| 79 | \( 1 - 9.15T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 8.24T + 89T^{2} \) |
| 97 | \( 1 - 2.18T + 97T^{2} \) |
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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
−8.345121360483923043900120446568, −7.87829602604869502154891104954, −6.98890758483376570007619622358, −6.64265791188034580056217829090, −5.09824761944150615794418044588, −4.99110172456217632628585428590, −4.30402067798938240792304427593, −3.29221799126032696049378261547, −1.64620270307687282364260057995, −0.52206988421321736225961631490,
0.52206988421321736225961631490, 1.64620270307687282364260057995, 3.29221799126032696049378261547, 4.30402067798938240792304427593, 4.99110172456217632628585428590, 5.09824761944150615794418044588, 6.64265791188034580056217829090, 6.98890758483376570007619622358, 7.87829602604869502154891104954, 8.345121360483923043900120446568
