| L(s) = 1 | + (−1.93 − 1.11i)5-s + (0.854 + 1.47i)7-s + (−3.21 + 1.85i)11-s + (6.85 − 11.8i)13-s + 15i·17-s − 14.4·19-s + (6.56 + 3.79i)23-s + (2.5 + 4.33i)25-s + (−8.40 + 4.85i)29-s + (10.2 − 17.6i)31-s − 3.81i·35-s − 56.2·37-s + (36.8 + 21.2i)41-s + (8.97 + 15.5i)43-s + (−1.22 + 0.708i)47-s + ⋯ |
| L(s) = 1 | + (−0.387 − 0.223i)5-s + (0.122 + 0.211i)7-s + (−0.291 + 0.168i)11-s + (0.527 − 0.913i)13-s + 0.882i·17-s − 0.758·19-s + (0.285 + 0.164i)23-s + (0.100 + 0.173i)25-s + (−0.289 + 0.167i)29-s + (0.329 − 0.570i)31-s − 0.109i·35-s − 1.52·37-s + (0.898 + 0.518i)41-s + (0.208 + 0.361i)43-s + (−0.0260 + 0.0150i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.047256298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.047256298\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.93 + 1.11i)T \) |
| good | 7 | \( 1 + (-0.854 - 1.47i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (3.21 - 1.85i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-6.85 + 11.8i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 15iT - 289T^{2} \) |
| 19 | \( 1 + 14.4T + 361T^{2} \) |
| 23 | \( 1 + (-6.56 - 3.79i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (8.40 - 4.85i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-10.2 + 17.6i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 56.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-36.8 - 21.2i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-8.97 - 15.5i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (1.22 - 0.708i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 70.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (26.7 + 15.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-40.7 - 70.5i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (7 - 12.1i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 1.95iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 27.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + (6.74 + 11.6i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (64.3 - 37.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 123. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-5.70 - 9.88i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.316366266354202974622786041936, −8.505416800516541270505002614738, −7.999292599894562712662208040009, −7.11075861895557333342489901994, −6.08630280199091223622600708468, −5.39995150480475146424786036074, −4.37222467370873858739778199983, −3.53578704230017208309227461240, −2.41818438558740635271946395241, −1.13431174764937880287901526630,
0.30691212912986200593499615800, 1.76703912983876248352806980957, 2.93867480382695305968408676982, 3.94084816874980068403640340605, 4.73011070733926339589101318254, 5.73489699773002039357552663460, 6.75786117859234112170175874680, 7.25166886768399497552783295963, 8.292197282090912694720831297458, 8.884161419871808111683571978706
