| L(s) = 1 | + 3-s + 3.42·5-s + 3.72·7-s + 9-s − 0.721·11-s − 2.12·13-s + 3.42·15-s − 0.126·17-s + 6.72·19-s + 3.72·21-s + 8.72·23-s + 6.72·25-s + 27-s + 0.721·29-s + 31-s − 0.721·33-s + 12.7·35-s + 7.42·37-s − 2.12·39-s − 7.44·41-s − 10.1·43-s + 3.42·45-s + 3.72·47-s + 6.84·49-s − 0.126·51-s − 7.29·53-s − 2.46·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.53·5-s + 1.40·7-s + 0.333·9-s − 0.217·11-s − 0.589·13-s + 0.883·15-s − 0.0305·17-s + 1.54·19-s + 0.812·21-s + 1.81·23-s + 1.34·25-s + 0.192·27-s + 0.133·29-s + 0.179·31-s − 0.125·33-s + 2.15·35-s + 1.22·37-s − 0.340·39-s − 1.16·41-s − 1.54·43-s + 0.510·45-s + 0.542·47-s + 0.978·49-s − 0.0176·51-s − 1.00·53-s − 0.332·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.686948443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.686948443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 3.42T + 5T^{2} \) |
| 7 | \( 1 - 3.72T + 7T^{2} \) |
| 11 | \( 1 + 0.721T + 11T^{2} \) |
| 13 | \( 1 + 2.12T + 13T^{2} \) |
| 17 | \( 1 + 0.126T + 17T^{2} \) |
| 19 | \( 1 - 6.72T + 19T^{2} \) |
| 23 | \( 1 - 8.72T + 23T^{2} \) |
| 29 | \( 1 - 0.721T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 7.42T + 37T^{2} \) |
| 41 | \( 1 + 7.44T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 3.72T + 47T^{2} \) |
| 53 | \( 1 + 7.29T + 53T^{2} \) |
| 59 | \( 1 - 0.828T + 59T^{2} \) |
| 61 | \( 1 + 9.19T + 61T^{2} \) |
| 67 | \( 1 - 3.17T + 67T^{2} \) |
| 71 | \( 1 + 5.31T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 9.44T + 79T^{2} \) |
| 83 | \( 1 - 5.85T + 83T^{2} \) |
| 89 | \( 1 - 7.97T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−7.76009014130266892351524876639, −7.29435955238848913902245488273, −6.47884050000989069704619480812, −5.56830726554803868333744167914, −5.01698134309669980778166582563, −4.61552492693973174824339365057, −3.19603404644527493949455262156, −2.64566904486375887852464876285, −1.70262546814340208075824819752, −1.19928533516231378532974307369,
1.19928533516231378532974307369, 1.70262546814340208075824819752, 2.64566904486375887852464876285, 3.19603404644527493949455262156, 4.61552492693973174824339365057, 5.01698134309669980778166582563, 5.56830726554803868333744167914, 6.47884050000989069704619480812, 7.29435955238848913902245488273, 7.76009014130266892351524876639
