| L(s) = 1 | − 2·2-s + 4·4-s + 7.28·5-s + 35.9·7-s − 8·8-s − 14.5·10-s + 41.0·11-s + 28.7·13-s − 71.9·14-s + 16·16-s + 1.14·17-s + 118.·19-s + 29.1·20-s − 82.1·22-s − 23·23-s − 71.9·25-s − 57.5·26-s + 143.·28-s − 274.·29-s − 93.9·31-s − 32·32-s − 2.29·34-s + 262.·35-s + 15.1·37-s − 237.·38-s − 58.2·40-s − 6.22·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.651·5-s + 1.94·7-s − 0.353·8-s − 0.460·10-s + 1.12·11-s + 0.613·13-s − 1.37·14-s + 0.250·16-s + 0.0163·17-s + 1.43·19-s + 0.325·20-s − 0.795·22-s − 0.208·23-s − 0.575·25-s − 0.433·26-s + 0.971·28-s − 1.75·29-s − 0.544·31-s − 0.176·32-s − 0.0115·34-s + 1.26·35-s + 0.0672·37-s − 1.01·38-s − 0.230·40-s − 0.0237·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.192542498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192542498\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 7.28T + 125T^{2} \) |
| 7 | \( 1 - 35.9T + 343T^{2} \) |
| 11 | \( 1 - 41.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 1.14T + 4.91e3T^{2} \) |
| 19 | \( 1 - 118.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 274.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 93.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 15.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.22T + 6.89e4T^{2} \) |
| 43 | \( 1 - 188.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 173.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 463.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 196.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 341.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 463.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 730.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 389.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 568.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 780.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 343.T + 9.12e5T^{2} \) |
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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
−11.03076703960772635669671189944, −9.681091949121643054439190331330, −9.069776445403156540306696369545, −8.051482744647273112023303739398, −7.35758280122472861280454882366, −6.03676909885752335465136893921, −5.14997088464339939817413146059, −3.78171341390933650005999085765, −1.93175907798387611102130871505, −1.24375995141890192343811577838,
1.24375995141890192343811577838, 1.93175907798387611102130871505, 3.78171341390933650005999085765, 5.14997088464339939817413146059, 6.03676909885752335465136893921, 7.35758280122472861280454882366, 8.051482744647273112023303739398, 9.069776445403156540306696369545, 9.681091949121643054439190331330, 11.03076703960772635669671189944
