| L(s) = 1 | − 2.17·2-s + 2.12·3-s + 2.71·4-s − 2.29·5-s − 4.62·6-s + 0.301·7-s − 1.54·8-s + 1.53·9-s + 4.98·10-s − 2.44·11-s + 5.77·12-s − 5.53·13-s − 0.655·14-s − 4.89·15-s − 2.06·16-s + 1.88·17-s − 3.32·18-s + 5.44·19-s − 6.23·20-s + 0.642·21-s + 5.31·22-s + 7.47·23-s − 3.29·24-s + 0.277·25-s + 12.0·26-s − 3.12·27-s + 0.819·28-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.22·3-s + 1.35·4-s − 1.02·5-s − 1.88·6-s + 0.114·7-s − 0.547·8-s + 0.510·9-s + 1.57·10-s − 0.738·11-s + 1.66·12-s − 1.53·13-s − 0.175·14-s − 1.26·15-s − 0.516·16-s + 0.458·17-s − 0.783·18-s + 1.24·19-s − 1.39·20-s + 0.140·21-s + 1.13·22-s + 1.55·23-s − 0.672·24-s + 0.0554·25-s + 2.35·26-s − 0.601·27-s + 0.154·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6755853202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6755853202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 3 | \( 1 - 2.12T + 3T^{2} \) |
| 5 | \( 1 + 2.29T + 5T^{2} \) |
| 7 | \( 1 - 0.301T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 5.53T + 13T^{2} \) |
| 17 | \( 1 - 1.88T + 17T^{2} \) |
| 19 | \( 1 - 5.44T + 19T^{2} \) |
| 23 | \( 1 - 7.47T + 23T^{2} \) |
| 29 | \( 1 + 7.37T + 29T^{2} \) |
| 31 | \( 1 + 8.92T + 31T^{2} \) |
| 37 | \( 1 - 2.38T + 37T^{2} \) |
| 41 | \( 1 + 1.71T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + 0.957T + 47T^{2} \) |
| 53 | \( 1 + 0.513T + 53T^{2} \) |
| 59 | \( 1 - 3.47T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 6.60T + 67T^{2} \) |
| 71 | \( 1 - 3.07T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 2.21T + 79T^{2} \) |
| 83 | \( 1 + 9.33T + 83T^{2} \) |
| 89 | \( 1 - 1.97T + 89T^{2} \) |
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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
−7.65596909651272253752018077549, −7.35172833180449452709311600132, −7.24419967336798869135698690138, −5.62891604451339879593719633050, −4.93154073347156836561867060424, −3.94151089771475776318104723076, −3.11380031135645963408832215868, −2.54334365317596449003018936505, −1.66339273497223544573028264656, −0.45797079937621577663820221508,
0.45797079937621577663820221508, 1.66339273497223544573028264656, 2.54334365317596449003018936505, 3.11380031135645963408832215868, 3.94151089771475776318104723076, 4.93154073347156836561867060424, 5.62891604451339879593719633050, 7.24419967336798869135698690138, 7.35172833180449452709311600132, 7.65596909651272253752018077549
