| L(s) = 1 | − 4.91·2-s + 16.1·4-s − 5·5-s − 39.8·8-s + 24.5·10-s − 12.0·11-s + 46.9·13-s + 66.8·16-s − 44.7·17-s − 31.7·19-s − 80.5·20-s + 58.9·22-s + 39.4·23-s + 25·25-s − 230.·26-s − 87.2·29-s + 304.·31-s − 9.27·32-s + 219.·34-s − 151.·37-s + 155.·38-s + 199.·40-s − 282.·41-s − 143.·43-s − 193.·44-s − 193.·46-s − 88.8·47-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.01·4-s − 0.447·5-s − 1.76·8-s + 0.776·10-s − 0.328·11-s + 1.00·13-s + 1.04·16-s − 0.637·17-s − 0.383·19-s − 0.900·20-s + 0.571·22-s + 0.357·23-s + 0.200·25-s − 1.73·26-s − 0.558·29-s + 1.76·31-s − 0.0512·32-s + 1.10·34-s − 0.671·37-s + 0.665·38-s + 0.787·40-s − 1.07·41-s − 0.507·43-s − 0.662·44-s − 0.620·46-s − 0.275·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.91T + 8T^{2} \) |
| 11 | \( 1 + 12.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 46.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 39.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 87.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 304.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 151.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 143.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 88.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 712.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 65.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 698.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 291.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 476.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 256.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.14e3T + 9.12e5T^{2} \) |
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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
−8.425571162290477388780433527223, −7.893326562427386680806748335807, −6.84366525560363363172482914472, −6.51213399151245181603647577998, −5.27759087917955294089130096293, −4.09907671594196140534705611449, −2.99886870079497091349977612725, −1.97366932482756549266064292955, −0.973373150579459016674029112394, 0,
0.973373150579459016674029112394, 1.97366932482756549266064292955, 2.99886870079497091349977612725, 4.09907671594196140534705611449, 5.27759087917955294089130096293, 6.51213399151245181603647577998, 6.84366525560363363172482914472, 7.893326562427386680806748335807, 8.425571162290477388780433527223
