| L(s) = 1 | − 1.58·3-s − 7.41·5-s + 34.2·7-s − 24.4·9-s + 56.3·11-s + 42.0·13-s + 11.7·15-s + 35.5·17-s − 46.8·19-s − 54.3·21-s + 134.·23-s − 69.9·25-s + 81.7·27-s − 159.·29-s + 159.·31-s − 89.5·33-s − 254.·35-s − 274.·37-s − 66.7·39-s + 265.·41-s − 242.·43-s + 181.·45-s − 67.3·47-s + 829.·49-s − 56.3·51-s + 215.·53-s − 418.·55-s + ⋯ |
| L(s) = 1 | − 0.305·3-s − 0.663·5-s + 1.84·7-s − 0.906·9-s + 1.54·11-s + 0.897·13-s + 0.202·15-s + 0.506·17-s − 0.565·19-s − 0.565·21-s + 1.21·23-s − 0.559·25-s + 0.582·27-s − 1.02·29-s + 0.923·31-s − 0.472·33-s − 1.22·35-s − 1.22·37-s − 0.274·39-s + 1.01·41-s − 0.861·43-s + 0.601·45-s − 0.209·47-s + 2.41·49-s − 0.154·51-s + 0.557·53-s − 1.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.272900443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.272900443\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 1.58T + 27T^{2} \) |
| 5 | \( 1 + 7.41T + 125T^{2} \) |
| 7 | \( 1 - 34.2T + 343T^{2} \) |
| 11 | \( 1 - 56.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 35.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 46.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 159.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 274.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 265.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 242.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 67.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 215.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 757.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 242.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.00e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 110.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 763.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 404.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 641.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 141.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 569.T + 9.12e5T^{2} \) |
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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
−9.352780119420086419966891851976, −8.534542318266317337985831741916, −8.119223769585768176276649084003, −7.09017228652573820709715387173, −6.09379358429462026691367773557, −5.18822791112966992356790947980, −4.29852654717914525842526646081, −3.43963623710195306036967031053, −1.82067984113248739700265150577, −0.872372754615033050371669927700,
0.872372754615033050371669927700, 1.82067984113248739700265150577, 3.43963623710195306036967031053, 4.29852654717914525842526646081, 5.18822791112966992356790947980, 6.09379358429462026691367773557, 7.09017228652573820709715387173, 8.119223769585768176276649084003, 8.534542318266317337985831741916, 9.352780119420086419966891851976
