| L(s) = 1 | − 2.58·2-s + 4.70·4-s + 1.14·5-s + 2.76·7-s − 6.99·8-s − 2.97·10-s + 4.35·11-s + 5.62·13-s − 7.15·14-s + 8.69·16-s − 2.41·17-s + 1.87·19-s + 5.40·20-s − 11.2·22-s + 8.82·23-s − 3.67·25-s − 14.5·26-s + 12.9·28-s + 1.56·31-s − 8.52·32-s + 6.25·34-s + 3.17·35-s + 7.76·37-s − 4.84·38-s − 8.03·40-s − 3.16·41-s − 1.46·43-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 2.35·4-s + 0.513·5-s + 1.04·7-s − 2.47·8-s − 0.940·10-s + 1.31·11-s + 1.56·13-s − 1.91·14-s + 2.17·16-s − 0.585·17-s + 0.429·19-s + 1.20·20-s − 2.40·22-s + 1.83·23-s − 0.735·25-s − 2.85·26-s + 2.45·28-s + 0.280·31-s − 1.50·32-s + 1.07·34-s + 0.537·35-s + 1.27·37-s − 0.786·38-s − 1.27·40-s − 0.493·41-s − 0.223·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.523218580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.523218580\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 5 | \( 1 - 1.14T + 5T^{2} \) |
| 7 | \( 1 - 2.76T + 7T^{2} \) |
| 11 | \( 1 - 4.35T + 11T^{2} \) |
| 13 | \( 1 - 5.62T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 - 1.87T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 31 | \( 1 - 1.56T + 31T^{2} \) |
| 37 | \( 1 - 7.76T + 37T^{2} \) |
| 41 | \( 1 + 3.16T + 41T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + 0.357T + 47T^{2} \) |
| 53 | \( 1 - 1.63T + 53T^{2} \) |
| 59 | \( 1 - 4.85T + 59T^{2} \) |
| 61 | \( 1 - 1.55T + 61T^{2} \) |
| 67 | \( 1 + 9.66T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 8.13T + 79T^{2} \) |
| 83 | \( 1 - 0.481T + 83T^{2} \) |
| 89 | \( 1 - 1.99T + 89T^{2} \) |
| 97 | \( 1 - 4.63T + 97T^{2} \) |
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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
−8.118866055724181330677463590423, −7.37517475227671280021571304417, −6.60902056198907690789750818426, −6.23398734153584285910798344838, −5.29937161230334962759542105389, −4.23778166775822782112252444248, −3.25603353608390205939643402296, −2.18830818436589585823428633647, −1.38579106697058137931520132756, −0.969138471386240969647744447329,
0.969138471386240969647744447329, 1.38579106697058137931520132756, 2.18830818436589585823428633647, 3.25603353608390205939643402296, 4.23778166775822782112252444248, 5.29937161230334962759542105389, 6.23398734153584285910798344838, 6.60902056198907690789750818426, 7.37517475227671280021571304417, 8.118866055724181330677463590423
