| L(s) = 1 | + (−0.5 + 1.65i)3-s + 3.31i·5-s + (−2.5 − 1.65i)9-s + 3.31i·11-s + (−5.5 − 1.65i)15-s − 3.31i·23-s − 6·25-s + (4 − 3.31i)27-s − 5·31-s + (−5.5 − 1.65i)33-s − 7·37-s + (5.5 − 8.29i)45-s + 6.63i·47-s + 7·49-s + 13.2i·53-s + ⋯ |
| L(s) = 1 | + (−0.288 + 0.957i)3-s + 1.48i·5-s + (−0.833 − 0.552i)9-s + 1.00i·11-s + (−1.42 − 0.428i)15-s − 0.691i·23-s − 1.20·25-s + (0.769 − 0.638i)27-s − 0.898·31-s + (−0.957 − 0.288i)33-s − 1.15·37-s + (0.819 − 1.23i)45-s + 0.967i·47-s + 49-s + 1.82i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.146168 + 0.991127i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.146168 + 0.991127i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 1.65i)T \) |
| 11 | \( 1 - 3.31iT \) |
| good | 5 | \( 1 - 3.31iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 3.31iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 6.63iT - 47T^{2} \) |
| 53 | \( 1 - 13.2iT - 53T^{2} \) |
| 59 | \( 1 + 3.31iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 - 16.5iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 16.5iT - 89T^{2} \) |
| 97 | \( 1 - 17T + 97T^{2} \) |
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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
−10.97935125414072199391846000303, −10.43604857520895397313552234020, −9.761033227496527930096902990677, −8.792118286725997175229462966309, −7.46155897196190805612781753676, −6.69743043306837242351934064057, −5.71868958286481369615025403707, −4.53277226051154901591940858948, −3.51364232222996827586422714352, −2.43257441357989869742041920563,
0.60389014896254339106856344580, 1.85953456650108989863298346040, 3.58599230319996879614848463687, 5.10269933282505063566711883265, 5.64440029760153032627169184873, 6.79421309109509854098817947383, 7.898670745427555371624584052664, 8.581527218580175625569397603300, 9.268737331075924840089156276012, 10.61967844044547947219519856922
