L(s) = 1 | + 2.23·3-s + 2.00·9-s + 2.23i·11-s − 4·13-s + 3i·17-s + 2.23i·19-s + 8.94i·23-s − 2.23·27-s + 4i·29-s − 8.94·31-s + 5.00i·33-s − 8·37-s − 8.94·39-s − 5·41-s + 8.94·43-s + ⋯ |
L(s) = 1 | + 1.29·3-s + 0.666·9-s + 0.674i·11-s − 1.10·13-s + 0.727i·17-s + 0.512i·19-s + 1.86i·23-s − 0.430·27-s + 0.742i·29-s − 1.60·31-s + 0.870i·33-s − 1.31·37-s − 1.43·39-s − 0.780·41-s + 1.36·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.887696175\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.887696175\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 2.23iT - 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 2.23iT - 19T^{2} \) |
| 23 | \( 1 - 8.94iT - 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 + 8.94iT - 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 8.94iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 9iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.967399757055391331899594815963, −8.157256670482435793450999866960, −7.35457121840572084177084298368, −7.12701773623691715929527247153, −5.71717624547148281109283547354, −5.08900498871730764880650832258, −3.86122537506813246325851626279, −3.45176798332281160174608292074, −2.27912459586941644552177631093, −1.68820591868850699911791856203,
0.43461534735373429331261795953, 2.12885921343181884274934325379, 2.68777077165942639177125906540, 3.51722055076621903200687976755, 4.45145378717671730327874574708, 5.27400105956159546481165973602, 6.28204832891751064218877376631, 7.24702850429936507875953315005, 7.67728130082023371983519313415, 8.637971423993784151168249027232