| L(s) = 1 | + 2-s + 4-s + 3.07·5-s − 0.254·7-s + 8-s + 3.07·10-s + 3.45·11-s + 5.63·13-s − 0.254·14-s + 16-s − 5.01·17-s − 0.904·19-s + 3.07·20-s + 3.45·22-s − 5.64·23-s + 4.48·25-s + 5.63·26-s − 0.254·28-s + 1.76·29-s − 10.6·31-s + 32-s − 5.01·34-s − 0.782·35-s + 8.83·37-s − 0.904·38-s + 3.07·40-s + 5.71·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.37·5-s − 0.0960·7-s + 0.353·8-s + 0.973·10-s + 1.04·11-s + 1.56·13-s − 0.0679·14-s + 0.250·16-s − 1.21·17-s − 0.207·19-s + 0.688·20-s + 0.736·22-s − 1.17·23-s + 0.896·25-s + 1.10·26-s − 0.0480·28-s + 0.328·29-s − 1.91·31-s + 0.176·32-s − 0.859·34-s − 0.132·35-s + 1.45·37-s − 0.146·38-s + 0.486·40-s + 0.892·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.566796768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.566796768\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 3.07T + 5T^{2} \) |
| 7 | \( 1 + 0.254T + 7T^{2} \) |
| 11 | \( 1 - 3.45T + 11T^{2} \) |
| 13 | \( 1 - 5.63T + 13T^{2} \) |
| 17 | \( 1 + 5.01T + 17T^{2} \) |
| 19 | \( 1 + 0.904T + 19T^{2} \) |
| 23 | \( 1 + 5.64T + 23T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 - 8.83T + 37T^{2} \) |
| 41 | \( 1 - 5.71T + 41T^{2} \) |
| 43 | \( 1 + 8.56T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 + 2.65T + 53T^{2} \) |
| 59 | \( 1 + 3.70T + 59T^{2} \) |
| 61 | \( 1 + 3.77T + 61T^{2} \) |
| 67 | \( 1 + 7.58T + 67T^{2} \) |
| 71 | \( 1 - 8.42T + 71T^{2} \) |
| 73 | \( 1 + 6.99T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 - 6.80T + 89T^{2} \) |
| 97 | \( 1 - 8.62T + 97T^{2} \) |
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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
−9.330530816707546341332478127847, −9.017514559908745837785478690177, −7.86026462805415473709235211822, −6.55002615264258232938342299858, −6.26719480662731757032940542428, −5.56902949111311163388566900234, −4.36184674668724263481837731894, −3.61949503758824306415126541247, −2.28161173126236378315224979125, −1.46650810120671451925436913613,
1.46650810120671451925436913613, 2.28161173126236378315224979125, 3.61949503758824306415126541247, 4.36184674668724263481837731894, 5.56902949111311163388566900234, 6.26719480662731757032940542428, 6.55002615264258232938342299858, 7.86026462805415473709235211822, 9.017514559908745837785478690177, 9.330530816707546341332478127847
