L(s) = 1 | + (−1.85 − 2.13i)2-s − 3.39i·3-s + (−1.08 + 7.92i)4-s − 21.1i·5-s + (−7.24 + 6.32i)6-s + 36.0·7-s + (18.9 − 12.4i)8-s + 15.4·9-s + (−45.0 + 39.3i)10-s − 1.32i·11-s + (26.9 + 3.67i)12-s − 22.9i·13-s + (−66.9 − 76.7i)14-s − 71.8·15-s + (−61.6 − 17.1i)16-s + 17·17-s + ⋯ |
L(s) = 1 | + (−0.657 − 0.753i)2-s − 0.654i·3-s + (−0.135 + 0.990i)4-s − 1.89i·5-s + (−0.492 + 0.430i)6-s + 1.94·7-s + (0.835 − 0.549i)8-s + 0.571·9-s + (−1.42 + 1.24i)10-s − 0.0362i·11-s + (0.648 + 0.0884i)12-s − 0.490i·13-s + (−1.27 − 1.46i)14-s − 1.23·15-s + (−0.963 − 0.267i)16-s + 0.242·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.418257 - 1.39655i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.418257 - 1.39655i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.85 + 2.13i)T \) |
| 17 | \( 1 - 17T \) |
good | 3 | \( 1 + 3.39iT - 27T^{2} \) |
| 5 | \( 1 + 21.1iT - 125T^{2} \) |
| 7 | \( 1 - 36.0T + 343T^{2} \) |
| 11 | \( 1 + 1.32iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 22.9iT - 2.19e3T^{2} \) |
| 19 | \( 1 - 50.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 44.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 74.4iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 160.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 371. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 264.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 144. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 445.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 243. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 404. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 474. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 424. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 54.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 648.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 157.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 523. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 603.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 142.T + 9.12e5T^{2} \) |
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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
−12.24635606268418008511338666795, −11.56124188748251000010920466828, −10.27699861987699807593477251659, −8.948746580680261155749918445112, −8.192698726887766601198391309899, −7.57391601464746703884786315849, −5.21555505095238635788645633325, −4.30288571985932738855201771137, −1.69002648042067349636920366650, −1.08554702412379586036751554806,
1.99125933702489408154862809801, 4.18770903929809594681747148313, 5.49429509524625898471911444247, 7.03892053443302369195910140239, 7.57085979158855837487400799271, 8.953396215506401824040388453249, 10.21309508502936398147428702216, 10.87281439097051642303629381097, 11.47992285494553693140188829568, 13.81031469478220251617922807935