| L(s) = 1 | − 2-s + 4-s − 3.79·5-s − 8-s + 3.79·10-s − 11-s + 0.361·13-s + 16-s + 4.11·17-s + 4.15·19-s − 3.79·20-s + 22-s − 0.542·23-s + 9.42·25-s − 0.361·26-s − 0.767·29-s − 8.80·31-s − 32-s − 4.11·34-s + 2.28·37-s − 4.15·38-s + 3.79·40-s − 9.13·41-s − 10.1·43-s − 44-s + 0.542·46-s + 2.98·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.69·5-s − 0.353·8-s + 1.20·10-s − 0.301·11-s + 0.100·13-s + 0.250·16-s + 0.998·17-s + 0.954·19-s − 0.849·20-s + 0.213·22-s − 0.113·23-s + 1.88·25-s − 0.0707·26-s − 0.142·29-s − 1.58·31-s − 0.176·32-s − 0.705·34-s + 0.375·37-s − 0.674·38-s + 0.600·40-s − 1.42·41-s − 1.55·43-s − 0.150·44-s + 0.0800·46-s + 0.435·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7036971129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7036971129\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3.79T + 5T^{2} \) |
| 13 | \( 1 - 0.361T + 13T^{2} \) |
| 17 | \( 1 - 4.11T + 17T^{2} \) |
| 19 | \( 1 - 4.15T + 19T^{2} \) |
| 23 | \( 1 + 0.542T + 23T^{2} \) |
| 29 | \( 1 + 0.767T + 29T^{2} \) |
| 31 | \( 1 + 8.80T + 31T^{2} \) |
| 37 | \( 1 - 2.28T + 37T^{2} \) |
| 41 | \( 1 + 9.13T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 2.98T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 2.25T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 0.603T + 67T^{2} \) |
| 71 | \( 1 - 6.87T + 71T^{2} \) |
| 73 | \( 1 - 9.45T + 73T^{2} \) |
| 79 | \( 1 - 0.0321T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 1.29T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−7.64639561792872286237024838392, −7.27778828988915799128135572717, −6.66564050193946044624245464395, −5.50361048404764323280807173334, −5.06207505048478503417029193598, −3.82658408903865063364265524386, −3.57831477886499631822968295562, −2.67332531852282836318390317425, −1.44543148355820861875495028477, −0.46830935610164087318279491479,
0.46830935610164087318279491479, 1.44543148355820861875495028477, 2.67332531852282836318390317425, 3.57831477886499631822968295562, 3.82658408903865063364265524386, 5.06207505048478503417029193598, 5.50361048404764323280807173334, 6.66564050193946044624245464395, 7.27778828988915799128135572717, 7.64639561792872286237024838392
