| L(s) = 1 | − 2.90·3-s − 2.20·5-s + 1.74·7-s + 5.41·9-s − 2.43·11-s − 2.67·13-s + 6.39·15-s + 5.15·17-s − 5.06·19-s − 5.07·21-s + 0.940·23-s − 0.144·25-s − 7.01·27-s + 1.06·29-s − 0.126·31-s + 7.06·33-s − 3.85·35-s + 9.61·37-s + 7.75·39-s − 7.00·41-s − 8.44·43-s − 11.9·45-s + 11.0·47-s − 3.94·49-s − 14.9·51-s + 2.69·53-s + 5.36·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s − 0.985·5-s + 0.660·7-s + 1.80·9-s − 0.734·11-s − 0.741·13-s + 1.65·15-s + 1.25·17-s − 1.16·19-s − 1.10·21-s + 0.196·23-s − 0.0288·25-s − 1.34·27-s + 0.198·29-s − 0.0226·31-s + 1.22·33-s − 0.651·35-s + 1.58·37-s + 1.24·39-s − 1.09·41-s − 1.28·43-s − 1.77·45-s + 1.61·47-s − 0.563·49-s − 2.09·51-s + 0.370·53-s + 0.723·55-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 | \( 1 + 2.90T + 3T^{2} \) |
| 5 | \( 1 + 2.20T + 5T^{2} \) |
| 7 | \( 1 - 1.74T + 7T^{2} \) |
| 11 | \( 1 + 2.43T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 - 5.15T + 17T^{2} \) |
| 19 | \( 1 + 5.06T + 19T^{2} \) |
| 23 | \( 1 - 0.940T + 23T^{2} \) |
| 29 | \( 1 - 1.06T + 29T^{2} \) |
| 31 | \( 1 + 0.126T + 31T^{2} \) |
| 37 | \( 1 - 9.61T + 37T^{2} \) |
| 41 | \( 1 + 7.00T + 41T^{2} \) |
| 43 | \( 1 + 8.44T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 4.57T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 1.50T + 73T^{2} \) |
| 79 | \( 1 - 3.05T + 79T^{2} \) |
| 83 | \( 1 + 8.08T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 9.42T + 97T^{2} \) |
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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
−7.46906360766073916863786345479, −6.79490489895142419605309688229, −5.96391387086441063158472567879, −5.38072962123131397776312503473, −4.71264584177391765056205888958, −4.27773852291593961932810766237, −3.23763762685703717093334401027, −2.03924491570386358446995002659, −0.875297644255531461280294734609, 0,
0.875297644255531461280294734609, 2.03924491570386358446995002659, 3.23763762685703717093334401027, 4.27773852291593961932810766237, 4.71264584177391765056205888958, 5.38072962123131397776312503473, 5.96391387086441063158472567879, 6.79490489895142419605309688229, 7.46906360766073916863786345479
