| L(s) = 1 | − 2.31·3-s + 0.927·5-s + 1.63·7-s + 2.34·9-s − 2.24·11-s − 4.71·13-s − 2.14·15-s − 2.14·17-s + 6.96·19-s − 3.78·21-s + 5.04·23-s − 4.13·25-s + 1.52·27-s − 5.17·29-s − 4.00·31-s + 5.18·33-s + 1.51·35-s − 0.244·37-s + 10.9·39-s + 7.08·41-s + 9.58·43-s + 2.17·45-s + 5.81·47-s − 4.31·49-s + 4.94·51-s + 2.15·53-s − 2.07·55-s + ⋯ |
| L(s) = 1 | − 1.33·3-s + 0.414·5-s + 0.619·7-s + 0.780·9-s − 0.675·11-s − 1.30·13-s − 0.553·15-s − 0.519·17-s + 1.59·19-s − 0.826·21-s + 1.05·23-s − 0.827·25-s + 0.292·27-s − 0.960·29-s − 0.718·31-s + 0.901·33-s + 0.256·35-s − 0.0401·37-s + 1.74·39-s + 1.10·41-s + 1.46·43-s + 0.323·45-s + 0.848·47-s − 0.616·49-s + 0.692·51-s + 0.295·53-s − 0.280·55-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 | \( 1 + 2.31T + 3T^{2} \) |
| 5 | \( 1 - 0.927T + 5T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 + 4.71T + 13T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 19 | \( 1 - 6.96T + 19T^{2} \) |
| 23 | \( 1 - 5.04T + 23T^{2} \) |
| 29 | \( 1 + 5.17T + 29T^{2} \) |
| 31 | \( 1 + 4.00T + 31T^{2} \) |
| 37 | \( 1 + 0.244T + 37T^{2} \) |
| 41 | \( 1 - 7.08T + 41T^{2} \) |
| 43 | \( 1 - 9.58T + 43T^{2} \) |
| 47 | \( 1 - 5.81T + 47T^{2} \) |
| 53 | \( 1 - 2.15T + 53T^{2} \) |
| 59 | \( 1 + 4.38T + 59T^{2} \) |
| 61 | \( 1 + 5.54T + 61T^{2} \) |
| 67 | \( 1 - 6.00T + 67T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 - 7.67T + 73T^{2} \) |
| 79 | \( 1 + 6.29T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−7.46647851277140367310333967051, −6.81659034969284761839663231152, −5.77412168735096105731138709321, −5.47790251693460396808898797945, −4.93195396925452443104257223903, −4.21409655693085240913428366383, −2.96317795520191482591856016738, −2.15282148247476354826662880692, −1.07220362467879091832477961900, 0,
1.07220362467879091832477961900, 2.15282148247476354826662880692, 2.96317795520191482591856016738, 4.21409655693085240913428366383, 4.93195396925452443104257223903, 5.47790251693460396808898797945, 5.77412168735096105731138709321, 6.81659034969284761839663231152, 7.46647851277140367310333967051
