L(s) = 1 | + 2-s − 1.36·3-s + 4-s + 0.363·5-s − 1.36·6-s + 2.14·7-s + 8-s − 1.14·9-s + 0.363·10-s − 1.36·12-s − 3.14·13-s + 2.14·14-s − 0.495·15-s + 16-s + 1.14·17-s − 1.14·18-s − 19-s + 0.363·20-s − 2.91·21-s − 2.14·23-s − 1.36·24-s − 4.86·25-s − 3.14·26-s + 5.64·27-s + 2.14·28-s − 1.77·29-s − 0.495·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.787·3-s + 0.5·4-s + 0.162·5-s − 0.556·6-s + 0.809·7-s + 0.353·8-s − 0.380·9-s + 0.114·10-s − 0.393·12-s − 0.871·13-s + 0.572·14-s − 0.127·15-s + 0.250·16-s + 0.276·17-s − 0.269·18-s − 0.229·19-s + 0.0812·20-s − 0.637·21-s − 0.446·23-s − 0.278·24-s − 0.973·25-s − 0.616·26-s + 1.08·27-s + 0.404·28-s − 0.330·29-s − 0.0904·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 1.36T + 3T^{2} \) |
| 5 | \( 1 - 0.363T + 5T^{2} \) |
| 7 | \( 1 - 2.14T + 7T^{2} \) |
| 13 | \( 1 + 3.14T + 13T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 29 | \( 1 + 1.77T + 29T^{2} \) |
| 31 | \( 1 - 7.00T + 31T^{2} \) |
| 37 | \( 1 + 9.77T + 37T^{2} \) |
| 41 | \( 1 + 7.91T + 41T^{2} \) |
| 43 | \( 1 + 4.28T + 43T^{2} \) |
| 47 | \( 1 + 7.59T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 1.08T + 59T^{2} \) |
| 61 | \( 1 + 8.51T + 61T^{2} \) |
| 67 | \( 1 - 8.56T + 67T^{2} \) |
| 71 | \( 1 - 3.50T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 0.778T + 79T^{2} \) |
| 83 | \( 1 + 0.778T + 83T^{2} \) |
| 89 | \( 1 + 8.20T + 89T^{2} \) |
| 97 | \( 1 + 1.86T + 97T^{2} \) |
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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
−8.025358327943759140158439447131, −6.94018233601734903851122908215, −6.45491054849334739547649896690, −5.45297460513828538521207937878, −5.20156982604335854050014148043, −4.41598122049440959941774048961, −3.44894263179504292792870571708, −2.42662095296569579447857421198, −1.53529571695812710730601693159, 0,
1.53529571695812710730601693159, 2.42662095296569579447857421198, 3.44894263179504292792870571708, 4.41598122049440959941774048961, 5.20156982604335854050014148043, 5.45297460513828538521207937878, 6.45491054849334739547649896690, 6.94018233601734903851122908215, 8.025358327943759140158439447131