L(s) = 1 | − 2.23·2-s + 3.00·4-s + (2.23 − 2.23i)7-s − 2.23·8-s + 3·9-s + (4 + 4i)11-s + (−4.47 + 4.47i)13-s + (−5.00 + 5.00i)14-s − 0.999·16-s + 4.47·17-s − 6.70·18-s + (2 − 2i)19-s + (−8.94 − 8.94i)22-s + (−2.23 − 2.23i)23-s + (10.0 − 10.0i)26-s + ⋯ |
L(s) = 1 | − 1.58·2-s + 1.50·4-s + (0.845 − 0.845i)7-s − 0.790·8-s + 9-s + (1.20 + 1.20i)11-s + (−1.24 + 1.24i)13-s + (−1.33 + 1.33i)14-s − 0.249·16-s + 1.08·17-s − 1.58·18-s + (0.458 − 0.458i)19-s + (−1.90 − 1.90i)22-s + (−0.466 − 0.466i)23-s + (1.96 − 1.96i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.849547 + 0.156049i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.849547 + 0.156049i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 + (2 - 5i)T \) |
good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + (-2.23 + 2.23i)T - 7iT^{2} \) |
| 11 | \( 1 + (-4 - 4i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.47 - 4.47i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + (-2 + 2i)T - 19iT^{2} \) |
| 23 | \( 1 + (2.23 + 2.23i)T + 23iT^{2} \) |
| 31 | \( 1 + (4 + 4i)T + 31iT^{2} \) |
| 37 | \( 1 - 4.47iT - 37T^{2} \) |
| 41 | \( 1 + (-1 + i)T - 41iT^{2} \) |
| 43 | \( 1 - 4.47iT - 43T^{2} \) |
| 47 | \( 1 + 4.47iT - 47T^{2} \) |
| 53 | \( 1 + (-8.94 - 8.94i)T + 53iT^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + (9 + 9i)T + 61iT^{2} \) |
| 67 | \( 1 + (-2.23 - 2.23i)T + 67iT^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 + (-2 + 2i)T - 79iT^{2} \) |
| 83 | \( 1 + (-6.70 - 6.70i)T + 83iT^{2} \) |
| 89 | \( 1 + (-7 + 7i)T - 89iT^{2} \) |
| 97 | \( 1 + 4.47iT - 97T^{2} \) |
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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
−10.03095069446887283500695752542, −9.697531610731478465954669893318, −8.967326600151403505434077369813, −7.66418491595981960244626955183, −7.29900167194767675144871450527, −6.71385539111449231587168571173, −4.78580527348481998391933332409, −4.12150757926765518937995571478, −1.98770239337389398298616446257, −1.24785424071002230282571312133,
0.942753274648310271344165429892, 2.06555761095676407786282457783, 3.57290454014270065124473920355, 5.14550684794159414929233760858, 6.06227702888113593816081160643, 7.41398140310428033778262602195, 7.81313517625536148431415805889, 8.698419388197151665336653787005, 9.477162418801577634192277542645, 10.11314811405201076562339695722