| L(s) = 1 | + (−1.83 − 0.667i)3-s + (0.302 + 1.71i)5-s + (0.962 − 1.66i)7-s + (0.622 + 0.522i)9-s + (2.27 + 3.94i)11-s + (2.57 − 0.938i)13-s + (0.589 − 3.34i)15-s + (3.84 − 3.22i)17-s + (4.18 − 1.21i)19-s + (−2.87 + 2.41i)21-s + (−0.198 + 1.12i)23-s + (1.85 − 0.675i)25-s + (2.13 + 3.69i)27-s + (3.83 + 3.21i)29-s + (0.999 − 1.73i)31-s + ⋯ |
| L(s) = 1 | + (−1.05 − 0.385i)3-s + (0.135 + 0.766i)5-s + (0.363 − 0.630i)7-s + (0.207 + 0.174i)9-s + (0.686 + 1.18i)11-s + (0.715 − 0.260i)13-s + (0.152 − 0.863i)15-s + (0.931 − 0.781i)17-s + (0.960 − 0.277i)19-s + (−0.628 + 0.527i)21-s + (−0.0412 + 0.234i)23-s + (0.371 − 0.135i)25-s + (0.411 + 0.711i)27-s + (0.712 + 0.597i)29-s + (0.179 − 0.311i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04133 - 0.0431718i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04133 - 0.0431718i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-4.18 + 1.21i)T \) |
| good | 3 | \( 1 + (1.83 + 0.667i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.302 - 1.71i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.962 + 1.66i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.27 - 3.94i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.57 + 0.938i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.84 + 3.22i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.198 - 1.12i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.83 - 3.21i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.999 + 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.06T + 37T^{2} \) |
| 41 | \( 1 + (10.3 + 3.77i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.05 + 6.01i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.72 - 5.63i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.00933 + 0.0529i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (2.83 - 2.37i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (2.60 - 14.7i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (9.72 + 8.15i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.359 - 2.03i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.670 - 0.243i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (10.5 + 3.85i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.99 + 8.64i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-15.2 + 5.54i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.285 + 0.239i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.89892171963082819636306790047, −10.75804513524619012498023959378, −10.20161146210119023894744782708, −8.960803025606187597059508230437, −7.34400684020214695116805183384, −6.97425010474428278055451997882, −5.83352856892043144316074750719, −4.77207084497470677292178298481, −3.26138886107790456393651957783, −1.24771311719264218875990778087,
1.21983041884585711715490480429, 3.49594923506599465448019297775, 4.92774281584044463676259841844, 5.66068323304762880874234688453, 6.43065931701615945816386886494, 8.239793694474117642705014191798, 8.775418832215165362495971829788, 9.999888773479019046755518591851, 10.94864565576926338171736738493, 11.80172462079028895543700204787
