| L(s) = 1 | + 2-s − 2.41·3-s + 4-s − 5-s − 2.41·6-s + 0.414·7-s + 8-s + 2.82·9-s − 10-s − 11-s − 2.41·12-s + 0.585·13-s + 0.414·14-s + 2.41·15-s + 16-s + 0.171·17-s + 2.82·18-s − 2.41·19-s − 20-s − 0.999·21-s − 22-s − 0.585·23-s − 2.41·24-s + 25-s + 0.585·26-s + 0.414·27-s + 0.414·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s − 0.447·5-s − 0.985·6-s + 0.156·7-s + 0.353·8-s + 0.942·9-s − 0.316·10-s − 0.301·11-s − 0.696·12-s + 0.162·13-s + 0.110·14-s + 0.623·15-s + 0.250·16-s + 0.0416·17-s + 0.666·18-s − 0.553·19-s − 0.223·20-s − 0.218·21-s − 0.213·22-s − 0.122·23-s − 0.492·24-s + 0.200·25-s + 0.114·26-s + 0.0797·27-s + 0.0782·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 7 | \( 1 - 0.414T + 7T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 - 0.171T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 + 0.585T + 23T^{2} \) |
| 29 | \( 1 + 0.414T + 29T^{2} \) |
| 31 | \( 1 - 3.82T + 31T^{2} \) |
| 37 | \( 1 + 0.757T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 47 | \( 1 + 7.65T + 47T^{2} \) |
| 53 | \( 1 + 4.17T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 + 8.07T + 61T^{2} \) |
| 67 | \( 1 - 7.65T + 67T^{2} \) |
| 71 | \( 1 + 7.24T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 + 2.24T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 1.58T + 89T^{2} \) |
| 97 | \( 1 + 5.17T + 97T^{2} \) |
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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
−7.84064242534832803047711173866, −6.82534417330357207323397614188, −6.44250368902191990297391884368, −5.63771781872608389005639413737, −5.01878403263890548998862716906, −4.41261936460111540379206733493, −3.56594820662870698799628768193, −2.49632115184157377782732432765, −1.24711858769399733439032519710, 0,
1.24711858769399733439032519710, 2.49632115184157377782732432765, 3.56594820662870698799628768193, 4.41261936460111540379206733493, 5.01878403263890548998862716906, 5.63771781872608389005639413737, 6.44250368902191990297391884368, 6.82534417330357207323397614188, 7.84064242534832803047711173866
