| L(s) = 1 | − 1.70·2-s − 5.10·4-s − 22.2·7-s + 22.2·8-s + 1.79·11-s − 58.2·13-s + 37.7·14-s + 2.89·16-s + 18.9·17-s + 104.·19-s − 3.04·22-s − 49.6·23-s + 99.0·26-s + 113.·28-s + 293.·29-s + 64.4·31-s − 183.·32-s − 32.3·34-s + 19.8·37-s − 178.·38-s + 165.·41-s + 247.·43-s − 9.14·44-s + 84.4·46-s + 384.·47-s + 150.·49-s + 297.·52-s + ⋯ |
| L(s) = 1 | − 0.601·2-s − 0.638·4-s − 1.19·7-s + 0.985·8-s + 0.0490·11-s − 1.24·13-s + 0.721·14-s + 0.0452·16-s + 0.270·17-s + 1.26·19-s − 0.0295·22-s − 0.449·23-s + 0.747·26-s + 0.765·28-s + 1.87·29-s + 0.373·31-s − 1.01·32-s − 0.162·34-s + 0.0883·37-s − 0.761·38-s + 0.630·41-s + 0.877·43-s − 0.0313·44-s + 0.270·46-s + 1.19·47-s + 0.438·49-s + 0.792·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7797505016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7797505016\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.70T + 8T^{2} \) |
| 7 | \( 1 + 22.2T + 343T^{2} \) |
| 11 | \( 1 - 1.79T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 18.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 293.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 64.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 19.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 165.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 247.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 384.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 463.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 73.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 173.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 594.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 320.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 770.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 173.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 384.T + 9.12e5T^{2} \) |
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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
−11.94903704569756358426326497767, −10.36240979570695940946407032855, −9.809125067411754901408620420159, −9.074810765468157147871308543604, −7.87282757374301840820356457019, −6.94352074344351613310366883575, −5.55077672619013929452988806514, −4.29877697518416051588241199469, −2.85016827522063193080890320155, −0.71535180307929917258243212967,
0.71535180307929917258243212967, 2.85016827522063193080890320155, 4.29877697518416051588241199469, 5.55077672619013929452988806514, 6.94352074344351613310366883575, 7.87282757374301840820356457019, 9.074810765468157147871308543604, 9.809125067411754901408620420159, 10.36240979570695940946407032855, 11.94903704569756358426326497767
