| L(s) = 1 | − 5.25·3-s − 9.15·7-s + 0.640·9-s − 11.8·11-s + 41.9·13-s + 75.7·17-s + 2.49·19-s + 48.1·21-s + 17.8·23-s + 138.·27-s − 143.·29-s − 88.6·31-s + 62.5·33-s + 351.·37-s − 220.·39-s + 195.·41-s − 366.·43-s − 58.5·47-s − 259.·49-s − 398.·51-s − 0.374·53-s − 13.0·57-s + 318.·59-s + 446.·61-s − 5.86·63-s + 709.·67-s − 93.8·69-s + ⋯ |
| L(s) = 1 | − 1.01·3-s − 0.494·7-s + 0.0237·9-s − 0.326·11-s + 0.895·13-s + 1.08·17-s + 0.0300·19-s + 0.500·21-s + 0.161·23-s + 0.987·27-s − 0.919·29-s − 0.513·31-s + 0.329·33-s + 1.55·37-s − 0.906·39-s + 0.746·41-s − 1.29·43-s − 0.181·47-s − 0.755·49-s − 1.09·51-s − 0.000971·53-s − 0.0304·57-s + 0.703·59-s + 0.937·61-s − 0.0117·63-s + 1.29·67-s − 0.163·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5.25T + 27T^{2} \) |
| 7 | \( 1 + 9.15T + 343T^{2} \) |
| 11 | \( 1 + 11.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 75.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.49T + 6.85e3T^{2} \) |
| 23 | \( 1 - 17.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 143.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 88.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 351.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 366.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 58.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 0.374T + 1.48e5T^{2} \) |
| 59 | \( 1 - 318.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 446.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 709.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 85.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 926.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 973.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3T + 9.12e5T^{2} \) |
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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
−9.633544254396041399754170146818, −8.596437350980618096506513542020, −7.67092005667711309083673738166, −6.62492099594118526435251016221, −5.84833862466170157586151412670, −5.24338579917369203923658798077, −3.95364981422226000725373511898, −2.88671134321781128301473437923, −1.22029348693172639227938317547, 0,
1.22029348693172639227938317547, 2.88671134321781128301473437923, 3.95364981422226000725373511898, 5.24338579917369203923658798077, 5.84833862466170157586151412670, 6.62492099594118526435251016221, 7.67092005667711309083673738166, 8.596437350980618096506513542020, 9.633544254396041399754170146818
