| L(s) = 1 | − 2.30·2-s − 2.16·3-s + 3.30·4-s + 2.16·5-s + 4.99·6-s − 3.00·8-s + 1.69·9-s − 4.99·10-s − 4.90·11-s − 7.15·12-s − 4.69·15-s + 0.302·16-s − 7.15·17-s − 3.90·18-s + 2.16·19-s + 7.15·20-s + 11.3·22-s − 0.605·23-s + 6.50·24-s − 0.302·25-s + 2.82·27-s − 2.30·29-s + 10.8·30-s − 7.15·31-s + 5.30·32-s + 10.6·33-s + 16.4·34-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 1.25·3-s + 1.65·4-s + 0.969·5-s + 2.03·6-s − 1.06·8-s + 0.565·9-s − 1.57·10-s − 1.47·11-s − 2.06·12-s − 1.21·15-s + 0.0756·16-s − 1.73·17-s − 0.921·18-s + 0.497·19-s + 1.60·20-s + 2.40·22-s − 0.126·23-s + 1.32·24-s − 0.0605·25-s + 0.543·27-s − 0.427·29-s + 1.97·30-s − 1.28·31-s + 0.937·32-s + 1.85·33-s + 2.82·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03590563729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03590563729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 + 2.16T + 3T^{2} \) |
| 5 | \( 1 - 2.16T + 5T^{2} \) |
| 11 | \( 1 + 4.90T + 11T^{2} \) |
| 17 | \( 1 + 7.15T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 + 0.605T + 23T^{2} \) |
| 29 | \( 1 + 2.30T + 29T^{2} \) |
| 31 | \( 1 + 7.15T + 31T^{2} \) |
| 37 | \( 1 + 8.60T + 37T^{2} \) |
| 41 | \( 1 + 9.98T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 - 2.39T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 4.33T + 61T^{2} \) |
| 67 | \( 1 + T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 4.33T + 73T^{2} \) |
| 79 | \( 1 + 6.60T + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 - 6.50T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−7.903267455033498800890602220895, −6.96841940003498918016422006985, −6.73307762548292579659942248142, −5.79740576475356885179277489876, −5.30550466669453878398302411373, −4.63648229161852666449257432764, −3.14054159572826942368238872582, −2.09441455474217205252611338083, −1.62356816974296849758901703186, −0.12367880362586509526980546787,
0.12367880362586509526980546787, 1.62356816974296849758901703186, 2.09441455474217205252611338083, 3.14054159572826942368238872582, 4.63648229161852666449257432764, 5.30550466669453878398302411373, 5.79740576475356885179277489876, 6.73307762548292579659942248142, 6.96841940003498918016422006985, 7.903267455033498800890602220895
