| L(s) = 1 | + 3·3-s + 0.128·5-s + 9·9-s + 54.0·11-s − 50.2·13-s + 0.385·15-s − 131.·17-s + 91.5·19-s + 179.·23-s − 124.·25-s + 27·27-s − 69.8·29-s + 326.·31-s + 162.·33-s + 301.·37-s − 150.·39-s + 296.·41-s − 144.·43-s + 1.15·45-s − 360.·47-s − 394.·51-s + 1.83·53-s + 6.94·55-s + 274.·57-s − 53.2·59-s − 108.·61-s − 6.45·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.0115·5-s + 0.333·9-s + 1.48·11-s − 1.07·13-s + 0.00664·15-s − 1.87·17-s + 1.10·19-s + 1.62·23-s − 0.999·25-s + 0.192·27-s − 0.447·29-s + 1.89·31-s + 0.854·33-s + 1.34·37-s − 0.618·39-s + 1.12·41-s − 0.511·43-s + 0.00383·45-s − 1.11·47-s − 1.08·51-s + 0.00475·53-s + 0.0170·55-s + 0.637·57-s − 0.117·59-s − 0.226·61-s − 0.0123·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.815928104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.815928104\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.128T + 125T^{2} \) |
| 11 | \( 1 - 54.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 91.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 179.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 69.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 326.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 296.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 144.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 360.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 1.83T + 1.48e5T^{2} \) |
| 59 | \( 1 + 53.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 108.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 842.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 241.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 206.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 559.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 986.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 443.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 740.T + 9.12e5T^{2} \) |
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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
−9.422439069827075740917076827939, −8.730445555686904601063308602989, −7.73658086029337434443099145934, −6.92352751613664542066789324421, −6.27145178491646226165658974094, −4.87895723984370030676337665858, −4.22382872682746669851071414461, −3.08336497180334901296730390957, −2.11565030303964816158125100604, −0.858338896293480329275752887957,
0.858338896293480329275752887957, 2.11565030303964816158125100604, 3.08336497180334901296730390957, 4.22382872682746669851071414461, 4.87895723984370030676337665858, 6.27145178491646226165658974094, 6.92352751613664542066789324421, 7.73658086029337434443099145934, 8.730445555686904601063308602989, 9.422439069827075740917076827939
