L(s) = 1 | + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−9.89 + 17.1i)5-s + 7.99·8-s + (−19.7 − 34.2i)10-s + (−7 − 12.1i)11-s + 50.9·13-s + (−8 + 13.8i)16-s + (0.707 + 1.22i)17-s + (0.707 − 1.22i)19-s + 79.1·20-s + 28·22-s + (70 − 121. i)23-s + (−133. − 231. i)25-s + (−50.9 + 88.1i)26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.885 + 1.53i)5-s + 0.353·8-s + (−0.626 − 1.08i)10-s + (−0.191 − 0.332i)11-s + 1.08·13-s + (−0.125 + 0.216i)16-s + (0.0100 + 0.0174i)17-s + (0.00853 − 0.0147i)19-s + 0.885·20-s + 0.271·22-s + (0.634 − 1.09i)23-s + (−1.06 − 1.84i)25-s + (−0.384 + 0.665i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.371150738\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.371150738\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (9.89 - 17.1i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (7 + 12.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 50.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-0.707 - 1.22i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-0.707 + 1.22i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-70 + 121. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 286T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-46.6 - 80.8i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-19 + 32.9i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 125.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 34T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-261. + 453. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (37 + 64.0i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-217. - 375. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (7.07 - 12.2i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (342 + 592. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 588T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-135. - 233. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (610 - 1.05e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 422.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-309. + 535. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−10.22985238940871350981732523822, −8.769570551381071282109700505025, −8.265801679928717862938038663824, −7.29044060457998333276473668418, −6.65514580025969549666061154320, −5.97519078801173983788063913493, −4.57921384408941327218223681188, −3.53652558586709856623007971661, −2.61534311121478685466171515051, −0.74287144411068182297460020405,
0.68548417447671785441001962924, 1.46293490880201235149352338711, 3.09464924440503464844772903846, 4.17333641762579906760814099454, 4.79747561396051656424404861612, 5.92221404011667504073193831164, 7.34568720709991810958393248464, 8.122097062248497637548550050058, 8.748572395178915060366304594953, 9.386592544806000823829766013020