| L(s) = 1 | − 4.92·3-s + 15.8·5-s − 17.8·7-s − 2.71·9-s + 52.9·11-s + 8.43·13-s − 78.1·15-s + 129.·17-s + 50.4·19-s + 87.9·21-s + 128.·23-s + 126.·25-s + 146.·27-s + 111.·29-s − 302.·31-s − 260.·33-s − 283.·35-s − 182.·37-s − 41.5·39-s − 94.5·41-s + 184.·43-s − 43.0·45-s + 296.·47-s − 24.1·49-s − 636.·51-s − 102.·53-s + 839.·55-s + ⋯ |
| L(s) = 1 | − 0.948·3-s + 1.41·5-s − 0.964·7-s − 0.100·9-s + 1.45·11-s + 0.179·13-s − 1.34·15-s + 1.84·17-s + 0.609·19-s + 0.914·21-s + 1.16·23-s + 1.01·25-s + 1.04·27-s + 0.712·29-s − 1.75·31-s − 1.37·33-s − 1.36·35-s − 0.813·37-s − 0.170·39-s − 0.360·41-s + 0.654·43-s − 0.142·45-s + 0.921·47-s − 0.0704·49-s − 1.74·51-s − 0.266·53-s + 2.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.469546360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.469546360\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.92T + 27T^{2} \) |
| 5 | \( 1 - 15.8T + 125T^{2} \) |
| 7 | \( 1 + 17.8T + 343T^{2} \) |
| 11 | \( 1 - 52.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.43T + 2.19e3T^{2} \) |
| 17 | \( 1 - 129.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 50.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 111.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 302.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 182.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 94.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 184.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 296.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 102.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 93.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 338.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 489.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 86.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 154.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 449.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 383.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 517.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.73e3T + 9.12e5T^{2} \) |
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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
−12.71294547861133488243939133042, −11.95218233620490209762578951165, −10.73529812629860444352909986815, −9.738122581831264796196262602920, −9.043305270717892583752902907929, −6.97979601724114239466939299649, −6.04956831398730683416393005564, −5.36196371718961953678413956835, −3.28791328557975573283912374552, −1.18789024419627928853200199750,
1.18789024419627928853200199750, 3.28791328557975573283912374552, 5.36196371718961953678413956835, 6.04956831398730683416393005564, 6.97979601724114239466939299649, 9.043305270717892583752902907929, 9.738122581831264796196262602920, 10.73529812629860444352909986815, 11.95218233620490209762578951165, 12.71294547861133488243939133042
