L(s) = 1 | + 2-s − 2.59·3-s + 4-s − 4.34·5-s − 2.59·6-s − 1.56·7-s + 8-s + 3.71·9-s − 4.34·10-s + 2.98·11-s − 2.59·12-s − 3.69·13-s − 1.56·14-s + 11.2·15-s + 16-s − 3.51·17-s + 3.71·18-s + 4.33·19-s − 4.34·20-s + 4.05·21-s + 2.98·22-s + 6.85·23-s − 2.59·24-s + 13.8·25-s − 3.69·26-s − 1.85·27-s − 1.56·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.49·3-s + 0.5·4-s − 1.94·5-s − 1.05·6-s − 0.590·7-s + 0.353·8-s + 1.23·9-s − 1.37·10-s + 0.899·11-s − 0.748·12-s − 1.02·13-s − 0.417·14-s + 2.90·15-s + 0.250·16-s − 0.852·17-s + 0.875·18-s + 0.994·19-s − 0.971·20-s + 0.883·21-s + 0.636·22-s + 1.42·23-s − 0.528·24-s + 2.77·25-s − 0.723·26-s − 0.357·27-s − 0.295·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
good | 3 | \( 1 + 2.59T + 3T^{2} \) |
| 5 | \( 1 + 4.34T + 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 13 | \( 1 + 3.69T + 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 - 4.33T + 19T^{2} \) |
| 23 | \( 1 - 6.85T + 23T^{2} \) |
| 29 | \( 1 + 8.88T + 29T^{2} \) |
| 31 | \( 1 - 0.941T + 31T^{2} \) |
| 37 | \( 1 - 7.68T + 37T^{2} \) |
| 41 | \( 1 + 7.33T + 41T^{2} \) |
| 43 | \( 1 - 7.74T + 43T^{2} \) |
| 47 | \( 1 - 8.67T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 + 8.43T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 0.318T + 67T^{2} \) |
| 71 | \( 1 + 6.87T + 71T^{2} \) |
| 73 | \( 1 + 4.25T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 7.86T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.60373045633886511956433317492, −7.15450975191221391290682237991, −6.69844076165958410423087508552, −5.76612521601908242205759969733, −4.94539553668676655013596557546, −4.39289404597926811412102485911, −3.70397332814339714825051773243, −2.82545513872696749166452448890, −1.00606571460926766530144788883, 0,
1.00606571460926766530144788883, 2.82545513872696749166452448890, 3.70397332814339714825051773243, 4.39289404597926811412102485911, 4.94539553668676655013596557546, 5.76612521601908242205759969733, 6.69844076165958410423087508552, 7.15450975191221391290682237991, 7.60373045633886511956433317492