| L(s) = 1 | + 1.17·2-s − 0.791·3-s − 0.624·4-s + 0.178·5-s − 0.928·6-s − 0.0809·7-s − 3.07·8-s − 2.37·9-s + 0.209·10-s − 4.30·11-s + 0.494·12-s + 1.10·13-s − 0.0948·14-s − 0.141·15-s − 2.35·16-s − 3.06·17-s − 2.78·18-s − 2.15·19-s − 0.111·20-s + 0.0640·21-s − 5.04·22-s + 9.15·23-s + 2.43·24-s − 4.96·25-s + 1.29·26-s + 4.25·27-s + 0.0505·28-s + ⋯ |
| L(s) = 1 | + 0.829·2-s − 0.457·3-s − 0.312·4-s + 0.0799·5-s − 0.379·6-s − 0.0305·7-s − 1.08·8-s − 0.791·9-s + 0.0662·10-s − 1.29·11-s + 0.142·12-s + 0.305·13-s − 0.0253·14-s − 0.0365·15-s − 0.589·16-s − 0.743·17-s − 0.655·18-s − 0.495·19-s − 0.0249·20-s + 0.0139·21-s − 1.07·22-s + 1.90·23-s + 0.497·24-s − 0.993·25-s + 0.253·26-s + 0.818·27-s + 0.00955·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 503 | \( 1 + T \) |
| good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 3 | \( 1 + 0.791T + 3T^{2} \) |
| 5 | \( 1 - 0.178T + 5T^{2} \) |
| 7 | \( 1 + 0.0809T + 7T^{2} \) |
| 11 | \( 1 + 4.30T + 11T^{2} \) |
| 13 | \( 1 - 1.10T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 + 2.15T + 19T^{2} \) |
| 23 | \( 1 - 9.15T + 23T^{2} \) |
| 29 | \( 1 + 6.17T + 29T^{2} \) |
| 31 | \( 1 + 9.82T + 31T^{2} \) |
| 37 | \( 1 - 4.08T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 - 2.03T + 43T^{2} \) |
| 47 | \( 1 + 5.12T + 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 5.62T + 67T^{2} \) |
| 71 | \( 1 + 4.88T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 + 2.48T + 83T^{2} \) |
| 89 | \( 1 - 5.57T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−10.96020347626974776631814317707, −9.493819482079649195599741751773, −8.786952821320320039862228805008, −7.72655579683521662917218541156, −6.43180590257067048186112748000, −5.52154731824581310271598715534, −4.97119951933508755479638605402, −3.69557217551358433867422098067, −2.54896388395685297495362863307, 0,
2.54896388395685297495362863307, 3.69557217551358433867422098067, 4.97119951933508755479638605402, 5.52154731824581310271598715534, 6.43180590257067048186112748000, 7.72655579683521662917218541156, 8.786952821320320039862228805008, 9.493819482079649195599741751773, 10.96020347626974776631814317707
