| L(s) = 1 | − 1.35·2-s + 0.880·3-s − 0.162·4-s + 5-s − 1.19·6-s − 3.54·7-s + 2.93·8-s − 2.22·9-s − 1.35·10-s − 6.04·11-s − 0.142·12-s − 0.780·13-s + 4.81·14-s + 0.880·15-s − 3.64·16-s − 3.48·17-s + 3.01·18-s − 4.59·19-s − 0.162·20-s − 3.12·21-s + 8.19·22-s + 2.96·23-s + 2.58·24-s + 25-s + 1.05·26-s − 4.59·27-s + 0.575·28-s + ⋯ |
| L(s) = 1 | − 0.958·2-s + 0.508·3-s − 0.0810·4-s + 0.447·5-s − 0.487·6-s − 1.34·7-s + 1.03·8-s − 0.741·9-s − 0.428·10-s − 1.82·11-s − 0.0412·12-s − 0.216·13-s + 1.28·14-s + 0.227·15-s − 0.912·16-s − 0.844·17-s + 0.710·18-s − 1.05·19-s − 0.0362·20-s − 0.681·21-s + 1.74·22-s + 0.619·23-s + 0.526·24-s + 0.200·25-s + 0.207·26-s − 0.885·27-s + 0.108·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2507566425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2507566425\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 3 | \( 1 - 0.880T + 3T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 + 6.04T + 11T^{2} \) |
| 13 | \( 1 + 0.780T + 13T^{2} \) |
| 17 | \( 1 + 3.48T + 17T^{2} \) |
| 19 | \( 1 + 4.59T + 19T^{2} \) |
| 23 | \( 1 - 2.96T + 23T^{2} \) |
| 29 | \( 1 - 5.27T + 29T^{2} \) |
| 37 | \( 1 + 4.63T + 37T^{2} \) |
| 41 | \( 1 + 7.86T + 41T^{2} \) |
| 43 | \( 1 - 0.825T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 9.04T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 1.81T + 61T^{2} \) |
| 67 | \( 1 + 7.06T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 - 1.06T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 3.40T + 89T^{2} \) |
| 97 | \( 1 + 5.51T + 97T^{2} \) |
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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
−8.472304535079587586798103479336, −7.85975606883891755581623996192, −6.90781242163075747140380259654, −6.35313913080618449420902668362, −5.30392541679114789270573352950, −4.70809712963102967872389005424, −3.44243117071066128575369892462, −2.72070290754976200306725100819, −2.00269884508904230506046073489, −0.28867199182917198821895248717,
0.28867199182917198821895248717, 2.00269884508904230506046073489, 2.72070290754976200306725100819, 3.44243117071066128575369892462, 4.70809712963102967872389005424, 5.30392541679114789270573352950, 6.35313913080618449420902668362, 6.90781242163075747140380259654, 7.85975606883891755581623996192, 8.472304535079587586798103479336
