| L(s) = 1 | − 3.55·3-s + 3.61·5-s − 2.20·7-s − 14.3·9-s − 44.9·11-s − 40.3·13-s − 12.8·15-s − 87.3·17-s + 47.5·19-s + 7.83·21-s − 60.0·23-s − 111.·25-s + 147.·27-s + 228.·29-s + 323.·31-s + 159.·33-s − 7.96·35-s − 315.·37-s + 143.·39-s + 133.·41-s − 117.·43-s − 51.8·45-s − 304.·47-s − 338.·49-s + 310.·51-s − 57.7·53-s − 162.·55-s + ⋯ |
| L(s) = 1 | − 0.684·3-s + 0.323·5-s − 0.118·7-s − 0.531·9-s − 1.23·11-s − 0.861·13-s − 0.221·15-s − 1.24·17-s + 0.574·19-s + 0.0814·21-s − 0.544·23-s − 0.895·25-s + 1.04·27-s + 1.46·29-s + 1.87·31-s + 0.842·33-s − 0.0384·35-s − 1.40·37-s + 0.589·39-s + 0.509·41-s − 0.415·43-s − 0.171·45-s − 0.946·47-s − 0.985·49-s + 0.853·51-s − 0.149·53-s − 0.398·55-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
| \(L(2)\) |
\(\approx\) |
\(0.7834072161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7834072161\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.55T + 27T^{2} \) |
| 5 | \( 1 - 3.61T + 125T^{2} \) |
| 7 | \( 1 + 2.20T + 343T^{2} \) |
| 11 | \( 1 + 44.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 40.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 87.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 47.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 60.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 228.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 323.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 315.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 133.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 117.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 304.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 57.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 450.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 401.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 342.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 608.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 582.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 305.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 115.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 940.T + 9.12e5T^{2} \) |
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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
−9.893213182517291185366478276754, −8.623623513016901278298184424950, −7.985421640095875763495277683508, −6.85296819468467138926285885139, −6.14202940278415828588771690053, −5.19766214455663319885670768365, −4.62301860688068559031319770474, −3.03025951487858807652772089283, −2.18280635235874042488452892965, −0.45764805775850117486906765228,
0.45764805775850117486906765228, 2.18280635235874042488452892965, 3.03025951487858807652772089283, 4.62301860688068559031319770474, 5.19766214455663319885670768365, 6.14202940278415828588771690053, 6.85296819468467138926285885139, 7.985421640095875763495277683508, 8.623623513016901278298184424950, 9.893213182517291185366478276754
