| L(s) = 1 | − 2.92·3-s − 1.53·5-s + 4.18·7-s + 5.58·9-s − 3.61·11-s + 1.36·13-s + 4.49·15-s + 2.36·17-s − 8.18·19-s − 12.2·21-s + 3.08·23-s − 2.64·25-s − 7.57·27-s + 5.68·29-s − 8.16·31-s + 10.6·33-s − 6.41·35-s + 1.96·37-s − 3.98·39-s + 9.60·41-s + 9.19·43-s − 8.56·45-s − 6.84·47-s + 10.5·49-s − 6.92·51-s − 5.91·53-s + 5.54·55-s + ⋯ |
| L(s) = 1 | − 1.69·3-s − 0.685·5-s + 1.58·7-s + 1.86·9-s − 1.09·11-s + 0.377·13-s + 1.15·15-s + 0.573·17-s − 1.87·19-s − 2.67·21-s + 0.643·23-s − 0.529·25-s − 1.45·27-s + 1.05·29-s − 1.46·31-s + 1.84·33-s − 1.08·35-s + 0.323·37-s − 0.638·39-s + 1.49·41-s + 1.40·43-s − 1.27·45-s − 0.997·47-s + 1.50·49-s − 0.969·51-s − 0.812·53-s + 0.747·55-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
| \(L(1)\) |
\(\approx\) |
\(0.8226761283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8226761283\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.92T + 3T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 - 4.18T + 7T^{2} \) |
| 11 | \( 1 + 3.61T + 11T^{2} \) |
| 13 | \( 1 - 1.36T + 13T^{2} \) |
| 17 | \( 1 - 2.36T + 17T^{2} \) |
| 19 | \( 1 + 8.18T + 19T^{2} \) |
| 23 | \( 1 - 3.08T + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 + 8.16T + 31T^{2} \) |
| 37 | \( 1 - 1.96T + 37T^{2} \) |
| 41 | \( 1 - 9.60T + 41T^{2} \) |
| 43 | \( 1 - 9.19T + 43T^{2} \) |
| 47 | \( 1 + 6.84T + 47T^{2} \) |
| 53 | \( 1 + 5.91T + 53T^{2} \) |
| 59 | \( 1 - 5.95T + 59T^{2} \) |
| 61 | \( 1 - 5.60T + 61T^{2} \) |
| 67 | \( 1 - 0.102T + 67T^{2} \) |
| 71 | \( 1 + 0.534T + 71T^{2} \) |
| 73 | \( 1 + 8.60T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 2.87T + 89T^{2} \) |
| 97 | \( 1 + 5.21T + 97T^{2} \) |
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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
−7.82515401811155209964555850077, −7.15738794887665518625016026307, −6.30138542288182613281866385360, −5.63770904342090341147360760624, −5.08520059076897728502439425276, −4.46971919536531902649721914311, −3.94318512992783253516632775849, −2.48348725251779498869030891024, −1.49379065467643381006761546385, −0.51724157665433537752669279547,
0.51724157665433537752669279547, 1.49379065467643381006761546385, 2.48348725251779498869030891024, 3.94318512992783253516632775849, 4.46971919536531902649721914311, 5.08520059076897728502439425276, 5.63770904342090341147360760624, 6.30138542288182613281866385360, 7.15738794887665518625016026307, 7.82515401811155209964555850077
