L(s) = 1 | + 2-s − 2.71·3-s + 4-s − 0.432·5-s − 2.71·6-s + 8-s + 4.34·9-s − 0.432·10-s − 4.46·11-s − 2.71·12-s − 6.22·13-s + 1.17·15-s + 16-s − 3.78·17-s + 4.34·18-s + 2.18·19-s − 0.432·20-s − 4.46·22-s − 0.639·23-s − 2.71·24-s − 4.81·25-s − 6.22·26-s − 3.65·27-s + 3.86·29-s + 1.17·30-s + 0.411·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.56·3-s + 0.5·4-s − 0.193·5-s − 1.10·6-s + 0.353·8-s + 1.44·9-s − 0.136·10-s − 1.34·11-s − 0.782·12-s − 1.72·13-s + 0.302·15-s + 0.250·16-s − 0.917·17-s + 1.02·18-s + 0.501·19-s − 0.0967·20-s − 0.951·22-s − 0.133·23-s − 0.553·24-s − 0.962·25-s − 1.22·26-s − 0.704·27-s + 0.718·29-s + 0.214·30-s + 0.0738·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8273108037\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8273108037\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + 2.71T + 3T^{2} \) |
| 5 | \( 1 + 0.432T + 5T^{2} \) |
| 11 | \( 1 + 4.46T + 11T^{2} \) |
| 13 | \( 1 + 6.22T + 13T^{2} \) |
| 17 | \( 1 + 3.78T + 17T^{2} \) |
| 19 | \( 1 - 2.18T + 19T^{2} \) |
| 23 | \( 1 + 0.639T + 23T^{2} \) |
| 29 | \( 1 - 3.86T + 29T^{2} \) |
| 31 | \( 1 - 0.411T + 31T^{2} \) |
| 37 | \( 1 - 7.45T + 37T^{2} \) |
| 43 | \( 1 - 2.67T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 5.53T + 67T^{2} \) |
| 71 | \( 1 + 9.40T + 71T^{2} \) |
| 73 | \( 1 - 6.26T + 73T^{2} \) |
| 79 | \( 1 + 0.703T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 1.59T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.093321580282230601036084586032, −7.47392474996511657289131602222, −6.83204491570017414698661590457, −6.04544470125185312587819187384, −5.36601337338183698204797645110, −4.82589635727517027818009582596, −4.29607697033074311237194309244, −2.92546274460253618429355753390, −2.10098141094549957932904451220, −0.47769402147953363434862378089,
0.47769402147953363434862378089, 2.10098141094549957932904451220, 2.92546274460253618429355753390, 4.29607697033074311237194309244, 4.82589635727517027818009582596, 5.36601337338183698204797645110, 6.04544470125185312587819187384, 6.83204491570017414698661590457, 7.47392474996511657289131602222, 8.093321580282230601036084586032