| L(s) = 1 | − 2.54·3-s + 2.86·5-s − 4.47·7-s + 3.45·9-s + 4.32·11-s − 6.10·13-s − 7.28·15-s + 3.82·17-s − 1.09·19-s + 11.3·21-s − 2.39·23-s + 3.21·25-s − 1.14·27-s − 3.71·31-s − 10.9·33-s − 12.8·35-s + 1.69·37-s + 15.5·39-s + 5.13·41-s − 2.00·43-s + 9.89·45-s − 9.28·47-s + 13.0·49-s − 9.70·51-s + 1.66·53-s + 12.3·55-s + 2.79·57-s + ⋯ |
| L(s) = 1 | − 1.46·3-s + 1.28·5-s − 1.69·7-s + 1.15·9-s + 1.30·11-s − 1.69·13-s − 1.88·15-s + 0.927·17-s − 0.252·19-s + 2.48·21-s − 0.498·23-s + 0.643·25-s − 0.220·27-s − 0.666·31-s − 1.91·33-s − 2.16·35-s + 0.277·37-s + 2.48·39-s + 0.801·41-s − 0.305·43-s + 1.47·45-s − 1.35·47-s + 1.86·49-s − 1.35·51-s + 0.229·53-s + 1.67·55-s + 0.369·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8841624552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8841624552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2.54T + 3T^{2} \) |
| 5 | \( 1 - 2.86T + 5T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 - 4.32T + 11T^{2} \) |
| 13 | \( 1 + 6.10T + 13T^{2} \) |
| 17 | \( 1 - 3.82T + 17T^{2} \) |
| 19 | \( 1 + 1.09T + 19T^{2} \) |
| 23 | \( 1 + 2.39T + 23T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 - 1.69T + 37T^{2} \) |
| 41 | \( 1 - 5.13T + 41T^{2} \) |
| 43 | \( 1 + 2.00T + 43T^{2} \) |
| 47 | \( 1 + 9.28T + 47T^{2} \) |
| 53 | \( 1 - 1.66T + 53T^{2} \) |
| 59 | \( 1 - 0.0901T + 59T^{2} \) |
| 61 | \( 1 - 8.52T + 61T^{2} \) |
| 67 | \( 1 + 8.77T + 67T^{2} \) |
| 71 | \( 1 - 9.23T + 71T^{2} \) |
| 73 | \( 1 - 2.10T + 73T^{2} \) |
| 79 | \( 1 - 5.42T + 79T^{2} \) |
| 83 | \( 1 + 6.23T + 83T^{2} \) |
| 89 | \( 1 + 3.62T + 89T^{2} \) |
| 97 | \( 1 - 6.64T + 97T^{2} \) |
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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
−9.012985559741126407320724744860, −7.53210742208176878762231083447, −6.68363130941481052388910357206, −6.38008158932146216004270864357, −5.71573406335475620562305462443, −5.12317868111834897608160373855, −4.06165568634943254846381034224, −2.99614549047729554833622527451, −1.88817865617250517774872943937, −0.58985551044840720427710827438,
0.58985551044840720427710827438, 1.88817865617250517774872943937, 2.99614549047729554833622527451, 4.06165568634943254846381034224, 5.12317868111834897608160373855, 5.71573406335475620562305462443, 6.38008158932146216004270864357, 6.68363130941481052388910357206, 7.53210742208176878762231083447, 9.012985559741126407320724744860
