| L(s) = 1 | + 0.112·3-s + 0.539·5-s + 1.01·7-s − 2.98·9-s − 3.54·11-s + 4.67·13-s + 0.0608·15-s + 1.84·17-s + 2.26·19-s + 0.114·21-s + 0.684·23-s − 4.70·25-s − 0.676·27-s + 1.75·29-s − 6.13·31-s − 0.400·33-s + 0.548·35-s − 8.12·37-s + 0.527·39-s + 11.9·41-s − 2.87·43-s − 1.61·45-s − 7.13·47-s − 5.96·49-s + 0.208·51-s − 5.70·53-s − 1.91·55-s + ⋯ |
| L(s) = 1 | + 0.0652·3-s + 0.241·5-s + 0.384·7-s − 0.995·9-s − 1.06·11-s + 1.29·13-s + 0.0157·15-s + 0.446·17-s + 0.519·19-s + 0.0250·21-s + 0.142·23-s − 0.941·25-s − 0.130·27-s + 0.324·29-s − 1.10·31-s − 0.0697·33-s + 0.0926·35-s − 1.33·37-s + 0.0845·39-s + 1.86·41-s − 0.439·43-s − 0.240·45-s − 1.04·47-s − 0.852·49-s + 0.0291·51-s − 0.784·53-s − 0.257·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 0.112T + 3T^{2} \) |
| 5 | \( 1 - 0.539T + 5T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 + 3.54T + 11T^{2} \) |
| 13 | \( 1 - 4.67T + 13T^{2} \) |
| 17 | \( 1 - 1.84T + 17T^{2} \) |
| 19 | \( 1 - 2.26T + 19T^{2} \) |
| 23 | \( 1 - 0.684T + 23T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 + 6.13T + 31T^{2} \) |
| 37 | \( 1 + 8.12T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 2.87T + 43T^{2} \) |
| 47 | \( 1 + 7.13T + 47T^{2} \) |
| 53 | \( 1 + 5.70T + 53T^{2} \) |
| 59 | \( 1 - 0.129T + 59T^{2} \) |
| 61 | \( 1 - 9.95T + 61T^{2} \) |
| 67 | \( 1 + 8.45T + 67T^{2} \) |
| 71 | \( 1 + 0.304T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 0.377T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 2.76T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−7.72514790812451617434775970392, −6.75843174545505240521160200741, −5.88655232740650806427346629928, −5.53051363336899478395188498648, −4.83112456528367826379481324477, −3.71869254965607396928950934593, −3.15151693280661246744975474909, −2.24436541387650105760799412674, −1.31841432151438831074266827219, 0,
1.31841432151438831074266827219, 2.24436541387650105760799412674, 3.15151693280661246744975474909, 3.71869254965607396928950934593, 4.83112456528367826379481324477, 5.53051363336899478395188498648, 5.88655232740650806427346629928, 6.75843174545505240521160200741, 7.72514790812451617434775970392
