| L(s) = 1 | + 2-s − 1.29·3-s + 4-s − 3.90·5-s − 1.29·6-s − 0.238·7-s + 8-s − 1.32·9-s − 3.90·10-s + 2.57·11-s − 1.29·12-s − 4.49·13-s − 0.238·14-s + 5.05·15-s + 16-s + 0.358·17-s − 1.32·18-s + 6.58·19-s − 3.90·20-s + 0.308·21-s + 2.57·22-s − 1.42·23-s − 1.29·24-s + 10.2·25-s − 4.49·26-s + 5.59·27-s − 0.238·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.747·3-s + 0.5·4-s − 1.74·5-s − 0.528·6-s − 0.0901·7-s + 0.353·8-s − 0.441·9-s − 1.23·10-s + 0.775·11-s − 0.373·12-s − 1.24·13-s − 0.0637·14-s + 1.30·15-s + 0.250·16-s + 0.0868·17-s − 0.312·18-s + 1.51·19-s − 0.873·20-s + 0.0674·21-s + 0.548·22-s − 0.297·23-s − 0.264·24-s + 2.05·25-s − 0.881·26-s + 1.07·27-s − 0.0450·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 1.29T + 3T^{2} \) |
| 5 | \( 1 + 3.90T + 5T^{2} \) |
| 7 | \( 1 + 0.238T + 7T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 - 0.358T + 17T^{2} \) |
| 19 | \( 1 - 6.58T + 19T^{2} \) |
| 23 | \( 1 + 1.42T + 23T^{2} \) |
| 29 | \( 1 - 6.25T + 29T^{2} \) |
| 31 | \( 1 - 3.78T + 31T^{2} \) |
| 37 | \( 1 + 4.29T + 37T^{2} \) |
| 41 | \( 1 + 3.63T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 7.73T + 47T^{2} \) |
| 53 | \( 1 - 6.47T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 7.71T + 61T^{2} \) |
| 67 | \( 1 - 3.45T + 67T^{2} \) |
| 71 | \( 1 + 1.94T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 4.85T + 79T^{2} \) |
| 83 | \( 1 - 2.51T + 83T^{2} \) |
| 89 | \( 1 - 4.10T + 89T^{2} \) |
| 97 | \( 1 - 5.69T + 97T^{2} \) |
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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
−7.64102018707704636445484003434, −6.83527516178544031902722492553, −6.47661558258244372087250191406, −5.18840353028689534386350814002, −5.01096678414831625930589118184, −4.08389891463972385553078931346, −3.38283279155924751729343850159, −2.70215960053148229451118397181, −1.08637669375570448352788053453, 0,
1.08637669375570448352788053453, 2.70215960053148229451118397181, 3.38283279155924751729343850159, 4.08389891463972385553078931346, 5.01096678414831625930589118184, 5.18840353028689534386350814002, 6.47661558258244372087250191406, 6.83527516178544031902722492553, 7.64102018707704636445484003434
