| L(s) = 1 | − 4.69·3-s − 7.54i·7-s + 13.0·9-s + (5.78 − 9.35i)11-s − 0.829i·13-s − 29.8i·17-s + 36.2i·19-s + 35.4i·21-s + 29.7·23-s − 18.9·27-s − 10.0i·29-s + 34.6·31-s + (−27.1 + 43.9i)33-s − 61.9·37-s + 3.89i·39-s + ⋯ |
| L(s) = 1 | − 1.56·3-s − 1.07i·7-s + 1.44·9-s + (0.525 − 0.850i)11-s − 0.0637i·13-s − 1.75i·17-s + 1.90i·19-s + 1.68i·21-s + 1.29·23-s − 0.700·27-s − 0.346i·29-s + 1.11·31-s + (−0.822 + 1.33i)33-s − 1.67·37-s + 0.0997i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8653703657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8653703657\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-5.78 + 9.35i)T \) |
| good | 3 | \( 1 + 4.69T + 9T^{2} \) |
| 7 | \( 1 + 7.54iT - 49T^{2} \) |
| 13 | \( 1 + 0.829iT - 169T^{2} \) |
| 17 | \( 1 + 29.8iT - 289T^{2} \) |
| 19 | \( 1 - 36.2iT - 361T^{2} \) |
| 23 | \( 1 - 29.7T + 529T^{2} \) |
| 29 | \( 1 + 10.0iT - 841T^{2} \) |
| 31 | \( 1 - 34.6T + 961T^{2} \) |
| 37 | \( 1 + 61.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 11.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 39.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 6.35T + 2.20e3T^{2} \) |
| 53 | \( 1 - 56.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 70.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 8.41iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 18.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 3.79T + 5.04e3T^{2} \) |
| 73 | \( 1 + 70.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 127. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 100. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 66.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + 1.65T + 9.40e3T^{2} \) |
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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
−9.687711536159455028896989099141, −8.543703881866263845305307031369, −7.39681819623388183791859022803, −6.80916703026657703295685763962, −5.94949171657269422601390317406, −5.17727229676631117423094316555, −4.29823473031366740261552296084, −3.24057200206102139143741734349, −1.23790800915057671172311032607, −0.42183609345929831538227576539,
1.14955536183045617835541152531, 2.46855708353109497526935209054, 4.06279301905813357249972220495, 5.06027545649981543940328063719, 5.55062835723574103154375152904, 6.71595029101118806899215004059, 6.88185451778919400877077503749, 8.484392535303262923884160903536, 9.107188732953002936536439669493, 10.15846195172817350876617748098
