| L(s) = 1 | − 8.72·3-s − 5·5-s − 28.9·7-s + 49.1·9-s − 6.25·11-s + 20.5·13-s + 43.6·15-s − 90.6·17-s − 100.·19-s + 252.·21-s + 23·23-s + 25·25-s − 193.·27-s + 95.0·29-s − 156.·31-s + 54.6·33-s + 144.·35-s − 25.0·37-s − 179.·39-s + 329.·41-s + 62.2·43-s − 245.·45-s + 356.·47-s + 492.·49-s + 791.·51-s − 465.·53-s + 31.2·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s − 0.447·5-s − 1.56·7-s + 1.82·9-s − 0.171·11-s + 0.438·13-s + 0.751·15-s − 1.29·17-s − 1.21·19-s + 2.62·21-s + 0.208·23-s + 0.200·25-s − 1.37·27-s + 0.608·29-s − 0.906·31-s + 0.288·33-s + 0.697·35-s − 0.111·37-s − 0.736·39-s + 1.25·41-s + 0.220·43-s − 0.814·45-s + 1.10·47-s + 1.43·49-s + 2.17·51-s − 1.20·53-s + 0.0767·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 8.72T + 27T^{2} \) |
| 7 | \( 1 + 28.9T + 343T^{2} \) |
| 11 | \( 1 + 6.25T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 90.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 100.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 95.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 25.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 329.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 62.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 356.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 465.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 246.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 312.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 706.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 834.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 723.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 553.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 102.T + 9.12e5T^{2} \) |
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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
−8.615511743031272747619831056110, −7.37086261578901406787004153025, −6.51671859624013383793076128584, −6.34929694737935457015768101744, −5.38606128234883879575715498573, −4.40907217148721828148755020184, −3.70398209911180902975110379916, −2.36167399613267921961668463334, −0.73509230947028949756311715863, 0,
0.73509230947028949756311715863, 2.36167399613267921961668463334, 3.70398209911180902975110379916, 4.40907217148721828148755020184, 5.38606128234883879575715498573, 6.34929694737935457015768101744, 6.51671859624013383793076128584, 7.37086261578901406787004153025, 8.615511743031272747619831056110
