| L(s) = 1 | − 5.58·2-s − 3·3-s + 23.1·4-s + 16.7·6-s + 10.0·7-s − 84.5·8-s + 9·9-s − 5.15·11-s − 69.4·12-s − 15.1·13-s − 56.0·14-s + 286.·16-s + 50.8·17-s − 50.2·18-s − 1.55·19-s − 30.1·21-s + 28.7·22-s − 187.·23-s + 253.·24-s + 84.4·26-s − 27·27-s + 232.·28-s − 29·29-s + 288.·31-s − 924.·32-s + 15.4·33-s − 283.·34-s + ⋯ |
| L(s) = 1 | − 1.97·2-s − 0.577·3-s + 2.89·4-s + 1.13·6-s + 0.541·7-s − 3.73·8-s + 0.333·9-s − 0.141·11-s − 1.67·12-s − 0.322·13-s − 1.06·14-s + 4.48·16-s + 0.725·17-s − 0.657·18-s − 0.0188·19-s − 0.312·21-s + 0.278·22-s − 1.70·23-s + 2.15·24-s + 0.636·26-s − 0.192·27-s + 1.56·28-s − 0.185·29-s + 1.66·31-s − 5.10·32-s + 0.0815·33-s − 1.43·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5300715546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5300715546\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 5.58T + 8T^{2} \) |
| 7 | \( 1 - 10.0T + 343T^{2} \) |
| 11 | \( 1 + 5.15T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 50.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 1.55T + 6.85e3T^{2} \) |
| 23 | \( 1 + 187.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 288.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 171.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 229.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 55.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 193.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 354.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 143.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 350.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 540.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 824.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 469.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 472.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 179.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3T + 9.12e5T^{2} \) |
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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
−8.604389846844277074381163316871, −8.018163657593143326916947157963, −7.49883693081091241844420372708, −6.55455180124857066532830854082, −5.98089078747578026899201812129, −4.95877991317201460049764387969, −3.46903230153238085588388802014, −2.28544979449810426100276865762, −1.48100851543201539151580972751, −0.47472420733111024207544476469,
0.47472420733111024207544476469, 1.48100851543201539151580972751, 2.28544979449810426100276865762, 3.46903230153238085588388802014, 4.95877991317201460049764387969, 5.98089078747578026899201812129, 6.55455180124857066532830854082, 7.49883693081091241844420372708, 8.018163657593143326916947157963, 8.604389846844277074381163316871
