| L(s) = 1 | + 3.18·2-s + 9.92·3-s + 2.17·4-s − 1.24·5-s + 31.6·6-s − 27.3·7-s − 18.5·8-s + 71.4·9-s − 3.95·10-s − 11·11-s + 21.5·12-s − 87.1·14-s − 12.3·15-s − 76.6·16-s + 72.7·17-s + 227.·18-s − 104.·19-s − 2.69·20-s − 271.·21-s − 35.0·22-s − 98.5·23-s − 184.·24-s − 123.·25-s + 441.·27-s − 59.3·28-s − 26.0·29-s − 39.2·30-s + ⋯ |
| L(s) = 1 | + 1.12·2-s + 1.90·3-s + 0.271·4-s − 0.110·5-s + 2.15·6-s − 1.47·7-s − 0.821·8-s + 2.64·9-s − 0.125·10-s − 0.301·11-s + 0.519·12-s − 1.66·14-s − 0.211·15-s − 1.19·16-s + 1.03·17-s + 2.98·18-s − 1.25·19-s − 0.0301·20-s − 2.81·21-s − 0.340·22-s − 0.893·23-s − 1.56·24-s − 0.987·25-s + 3.14·27-s − 0.400·28-s − 0.166·29-s − 0.238·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3.18T + 8T^{2} \) |
| 3 | \( 1 - 9.92T + 27T^{2} \) |
| 5 | \( 1 + 1.24T + 125T^{2} \) |
| 7 | \( 1 + 27.3T + 343T^{2} \) |
| 17 | \( 1 - 72.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 98.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 26.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 170.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 340.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 261.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 10.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 577.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 2.47T + 1.48e5T^{2} \) |
| 59 | \( 1 - 109.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 678.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 320.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 502.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 310.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 410.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 809.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 993.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 172.T + 9.12e5T^{2} \) |
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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
−8.436981369866148756181981664121, −7.85768577116099177730752220310, −6.76216587290631287374637543259, −6.19290853101676344433890995480, −4.94632586030325510151073191030, −3.91702601083358788548991065690, −3.49798400302828978822848164586, −2.85195230196389376411278775702, −1.91727274551041571068685054482, 0,
1.91727274551041571068685054482, 2.85195230196389376411278775702, 3.49798400302828978822848164586, 3.91702601083358788548991065690, 4.94632586030325510151073191030, 6.19290853101676344433890995480, 6.76216587290631287374637543259, 7.85768577116099177730752220310, 8.436981369866148756181981664121
