L(s) = 1 | − 5.52·3-s + 68.9·7-s − 212.·9-s + 486.·11-s + 428.·13-s − 1.80e3·17-s + 1.04e3·19-s − 380.·21-s + 686.·23-s + 2.51e3·27-s − 1.33e3·29-s − 7.99e3·31-s − 2.68e3·33-s + 1.97e3·37-s − 2.36e3·39-s + 1.07e4·41-s + 1.50e4·43-s + 895.·47-s − 1.20e4·49-s + 9.94e3·51-s + 1.93e4·53-s − 5.78e3·57-s − 2.11e4·59-s − 2.77e4·61-s − 1.46e4·63-s − 7.71e3·67-s − 3.79e3·69-s + ⋯ |
L(s) = 1 | − 0.354·3-s + 0.531·7-s − 0.874·9-s + 1.21·11-s + 0.703·13-s − 1.51·17-s + 0.665·19-s − 0.188·21-s + 0.270·23-s + 0.664·27-s − 0.295·29-s − 1.49·31-s − 0.429·33-s + 0.236·37-s − 0.249·39-s + 1.00·41-s + 1.23·43-s + 0.0591·47-s − 0.717·49-s + 0.535·51-s + 0.945·53-s − 0.235·57-s − 0.792·59-s − 0.953·61-s − 0.465·63-s − 0.210·67-s − 0.0959·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.866688079\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.866688079\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 + 5.52T + 243T^{2} \) |
| 7 | \( 1 - 68.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 486.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 428.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.80e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 686.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.33e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.97e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.50e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 895.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.77e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.71e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.22e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.48e5T + 8.58e9T^{2} \) |
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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
−10.89076753273590244511245547042, −9.238665763092530542728124660557, −8.867865931595605439979241255843, −7.65835145371160270939437195432, −6.54663478839393067463603638661, −5.75541962469641083996100860404, −4.60557277768985972496482298225, −3.52202524990560935427513190270, −2.04214349764026624114689933067, −0.74531022172690809737111640214,
0.74531022172690809737111640214, 2.04214349764026624114689933067, 3.52202524990560935427513190270, 4.60557277768985972496482298225, 5.75541962469641083996100860404, 6.54663478839393067463603638661, 7.65835145371160270939437195432, 8.867865931595605439979241255843, 9.238665763092530542728124660557, 10.89076753273590244511245547042