| L(s) = 1 | − 3.16·3-s − 0.445·5-s − 3.10·7-s + 7.02·9-s − 3.11·11-s + 0.671·13-s + 1.41·15-s + 3.69·17-s + 5.95·19-s + 9.83·21-s + 1.71·23-s − 4.80·25-s − 12.7·27-s − 10.0·29-s − 9.96·31-s + 9.86·33-s + 1.38·35-s + 0.690·37-s − 2.12·39-s − 9.97·41-s + 8.69·43-s − 3.12·45-s + 1.00·47-s + 2.66·49-s − 11.7·51-s + 7.55·53-s + 1.38·55-s + ⋯ |
| L(s) = 1 | − 1.82·3-s − 0.199·5-s − 1.17·7-s + 2.34·9-s − 0.939·11-s + 0.186·13-s + 0.364·15-s + 0.896·17-s + 1.36·19-s + 2.14·21-s + 0.358·23-s − 0.960·25-s − 2.44·27-s − 1.86·29-s − 1.78·31-s + 1.71·33-s + 0.234·35-s + 0.113·37-s − 0.340·39-s − 1.55·41-s + 1.32·43-s − 0.466·45-s + 0.147·47-s + 0.380·49-s − 1.63·51-s + 1.03·53-s + 0.187·55-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 | \( 1 + 3.16T + 3T^{2} \) |
| 5 | \( 1 + 0.445T + 5T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 - 0.671T + 13T^{2} \) |
| 17 | \( 1 - 3.69T + 17T^{2} \) |
| 19 | \( 1 - 5.95T + 19T^{2} \) |
| 23 | \( 1 - 1.71T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 9.96T + 31T^{2} \) |
| 37 | \( 1 - 0.690T + 37T^{2} \) |
| 41 | \( 1 + 9.97T + 41T^{2} \) |
| 43 | \( 1 - 8.69T + 43T^{2} \) |
| 47 | \( 1 - 1.00T + 47T^{2} \) |
| 53 | \( 1 - 7.55T + 53T^{2} \) |
| 59 | \( 1 - 5.09T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 2.85T + 67T^{2} \) |
| 71 | \( 1 - 6.00T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 2.25T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.36267916277209048573160990349, −6.76949037580170661059065619190, −5.79760122672668535942457093943, −5.57557847836618518060598360953, −5.04737602227592907895538754287, −3.80745426669272532866317306159, −3.43872266332092744481291846537, −2.03985188609881658840506810414, −0.832006418136945965493493660134, 0,
0.832006418136945965493493660134, 2.03985188609881658840506810414, 3.43872266332092744481291846537, 3.80745426669272532866317306159, 5.04737602227592907895538754287, 5.57557847836618518060598360953, 5.79760122672668535942457093943, 6.76949037580170661059065619190, 7.36267916277209048573160990349
