L(s) = 1 | + 2-s + 1.59·3-s + 4-s − 0.438·5-s + 1.59·6-s − 0.361·7-s + 8-s − 0.458·9-s − 0.438·10-s + 4.42·11-s + 1.59·12-s + 4.06·13-s − 0.361·14-s − 0.698·15-s + 16-s + 4.13·17-s − 0.458·18-s + 2.03·19-s − 0.438·20-s − 0.576·21-s + 4.42·22-s + 23-s + 1.59·24-s − 4.80·25-s + 4.06·26-s − 5.51·27-s − 0.361·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.920·3-s + 0.5·4-s − 0.195·5-s + 0.650·6-s − 0.136·7-s + 0.353·8-s − 0.152·9-s − 0.138·10-s + 1.33·11-s + 0.460·12-s + 1.12·13-s − 0.0967·14-s − 0.180·15-s + 0.250·16-s + 1.00·17-s − 0.108·18-s + 0.466·19-s − 0.0979·20-s − 0.125·21-s + 0.942·22-s + 0.208·23-s + 0.325·24-s − 0.961·25-s + 0.796·26-s − 1.06·27-s − 0.0683·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.965454507\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.965454507\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 - 1.59T + 3T^{2} \) |
| 5 | \( 1 + 0.438T + 5T^{2} \) |
| 7 | \( 1 + 0.361T + 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 13 | \( 1 - 4.06T + 13T^{2} \) |
| 17 | \( 1 - 4.13T + 17T^{2} \) |
| 19 | \( 1 - 2.03T + 19T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 - 8.21T + 31T^{2} \) |
| 37 | \( 1 - 5.31T + 37T^{2} \) |
| 41 | \( 1 + 9.41T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 0.939T + 47T^{2} \) |
| 53 | \( 1 + 1.19T + 53T^{2} \) |
| 59 | \( 1 - 5.01T + 59T^{2} \) |
| 61 | \( 1 + 1.04T + 61T^{2} \) |
| 67 | \( 1 + 1.18T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 1.86T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 8.29T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−8.042357852378170171898382265531, −7.46788493596704472816937265979, −6.50791027908409077886726297892, −5.99447247037841802150881175199, −5.23362807834901937712629511438, −4.06069800930639333593166573737, −3.66608430006776373044834916554, −3.07727303323766120090766069208, −2.01477184853742238094889174307, −1.08182591585893617082808592636,
1.08182591585893617082808592636, 2.01477184853742238094889174307, 3.07727303323766120090766069208, 3.66608430006776373044834916554, 4.06069800930639333593166573737, 5.23362807834901937712629511438, 5.99447247037841802150881175199, 6.50791027908409077886726297892, 7.46788493596704472816937265979, 8.042357852378170171898382265531