| L(s) = 1 | + 1.33·3-s + 3.45·5-s − 1.09·7-s − 1.20·9-s − 2.07·11-s − 0.787·13-s + 4.61·15-s − 17-s + 5.10·19-s − 1.46·21-s + 5.70·23-s + 6.90·25-s − 5.63·27-s + 0.628·29-s + 11.0·31-s − 2.77·33-s − 3.78·35-s + 1.84·37-s − 1.05·39-s + 3.21·41-s − 11.9·43-s − 4.17·45-s + 7.01·47-s − 5.79·49-s − 1.33·51-s + 7.97·53-s − 7.14·55-s + ⋯ |
| L(s) = 1 | + 0.772·3-s + 1.54·5-s − 0.414·7-s − 0.403·9-s − 0.624·11-s − 0.218·13-s + 1.19·15-s − 0.242·17-s + 1.17·19-s − 0.320·21-s + 1.18·23-s + 1.38·25-s − 1.08·27-s + 0.116·29-s + 1.97·31-s − 0.482·33-s − 0.640·35-s + 0.302·37-s − 0.168·39-s + 0.502·41-s − 1.82·43-s − 0.621·45-s + 1.02·47-s − 0.827·49-s − 0.187·51-s + 1.09·53-s − 0.963·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.157471039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.157471039\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 1.33T + 3T^{2} \) |
| 5 | \( 1 - 3.45T + 5T^{2} \) |
| 7 | \( 1 + 1.09T + 7T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 + 0.787T + 13T^{2} \) |
| 19 | \( 1 - 5.10T + 19T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 - 0.628T + 29T^{2} \) |
| 31 | \( 1 - 11.0T + 31T^{2} \) |
| 37 | \( 1 - 1.84T + 37T^{2} \) |
| 41 | \( 1 - 3.21T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 - 7.01T + 47T^{2} \) |
| 53 | \( 1 - 7.97T + 53T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 0.837T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 7.79T + 79T^{2} \) |
| 83 | \( 1 + 1.69T + 83T^{2} \) |
| 89 | \( 1 - 3.60T + 89T^{2} \) |
| 97 | \( 1 - 9.09T + 97T^{2} \) |
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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
−8.529269642050583443356407821050, −7.86314843945461177410920324019, −6.87489618508247136382008792278, −6.28477612599322014001608004142, −5.38361532157515344109441146220, −4.96397456075002837735345419244, −3.54410275120406439839095462832, −2.71861106838039716481526841769, −2.30830505425311793208094531610, −1.00364702683528751090831781576,
1.00364702683528751090831781576, 2.30830505425311793208094531610, 2.71861106838039716481526841769, 3.54410275120406439839095462832, 4.96397456075002837735345419244, 5.38361532157515344109441146220, 6.28477612599322014001608004142, 6.87489618508247136382008792278, 7.86314843945461177410920324019, 8.529269642050583443356407821050
