| L(s) = 1 | + 5.10·2-s + 5.20·3-s + 18.0·4-s − 5.29·5-s + 26.5·6-s + 51.0·8-s + 0.0621·9-s − 26.9·10-s + 53.8·11-s + 93.7·12-s + 27.6·13-s − 27.5·15-s + 116.·16-s + 100.·17-s + 0.317·18-s + 70.0·19-s − 95.3·20-s + 274.·22-s − 147.·23-s + 265.·24-s − 97.0·25-s + 140.·26-s − 140.·27-s + 78.3·29-s − 140.·30-s − 79.8·31-s + 185.·32-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 1.00·3-s + 2.25·4-s − 0.473·5-s + 1.80·6-s + 2.25·8-s + 0.00230·9-s − 0.853·10-s + 1.47·11-s + 2.25·12-s + 0.588·13-s − 0.473·15-s + 1.81·16-s + 1.43·17-s + 0.00415·18-s + 0.845·19-s − 1.06·20-s + 2.66·22-s − 1.33·23-s + 2.26·24-s − 0.776·25-s + 1.06·26-s − 0.998·27-s + 0.501·29-s − 0.854·30-s − 0.462·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.58396085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.58396085\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 5.10T + 8T^{2} \) |
| 3 | \( 1 - 5.20T + 27T^{2} \) |
| 5 | \( 1 + 5.29T + 125T^{2} \) |
| 11 | \( 1 - 53.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 100.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 70.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 78.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 79.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 245.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 308.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 363.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 264.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 195.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 716.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 478.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 129.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 459.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 392.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 581.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 935.T + 9.12e5T^{2} \) |
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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
−8.354230300513571293251938660337, −7.71884903463934993779193044551, −6.95334110778229091609790967576, −5.97815993267730950196813455353, −5.54758132658464378693288958477, −4.20072873630948182481435141642, −3.79875828299333202367706068725, −3.22998818951958231941918257155, −2.25634827001679425147351517473, −1.19798559486011033307996294660,
1.19798559486011033307996294660, 2.25634827001679425147351517473, 3.22998818951958231941918257155, 3.79875828299333202367706068725, 4.20072873630948182481435141642, 5.54758132658464378693288958477, 5.97815993267730950196813455353, 6.95334110778229091609790967576, 7.71884903463934993779193044551, 8.354230300513571293251938660337
