L(s) = 1 | + (−2 + 3.46i)5-s + (−10 − 17.3i)11-s + 4·13-s + (−12 − 20.7i)17-s + (22 − 38.1i)19-s + (36 − 62.3i)23-s + (54.5 + 94.3i)25-s + 38·29-s + (92 + 159. i)31-s + (15 − 25.9i)37-s − 216·41-s − 164·43-s + (−260 + 450. i)47-s + (−73 − 126. i)53-s + 80·55-s + ⋯ |
L(s) = 1 | + (−0.178 + 0.309i)5-s + (−0.274 − 0.474i)11-s + 0.0853·13-s + (−0.171 − 0.296i)17-s + (0.265 − 0.460i)19-s + (0.326 − 0.565i)23-s + (0.435 + 0.755i)25-s + 0.243·29-s + (0.533 + 0.923i)31-s + (0.0666 − 0.115i)37-s − 0.822·41-s − 0.581·43-s + (−0.806 + 1.39i)47-s + (−0.189 − 0.327i)53-s + 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8014906983\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8014906983\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (2 - 3.46i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (10 + 17.3i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 4T + 2.19e3T^{2} \) |
| 17 | \( 1 + (12 + 20.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-22 + 38.1i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-36 + 62.3i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 38T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-92 - 159. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-15 + 25.9i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 216T + 6.89e4T^{2} \) |
| 43 | \( 1 + 164T + 7.95e4T^{2} \) |
| 47 | \( 1 + (260 - 450. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (73 + 126. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (230 + 398. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-314 + 543. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (278 + 481. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 592T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-512 - 886. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-52 + 90.0i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 324T + 5.71e5T^{2} \) |
| 89 | \( 1 + (448 - 775. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 920T + 9.12e5T^{2} \) |
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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
−8.608631575417185464697964357267, −7.934811107406342843475578495141, −6.96639465165704770090151683921, −6.42678431701514713838670882063, −5.29562113833009318241056819934, −4.63870743234786396276210762386, −3.39561290030609142901828588870, −2.77629112207414538387656546430, −1.41719066393844965239040771789, −0.18302262650158971970537391434,
1.11176387415633911350809346862, 2.23156081553992269578077556720, 3.33829678064035108361884927186, 4.32135133891677080351519333796, 5.07784735614588669937628326463, 6.00289942750689493643912683824, 6.86419916446445564615308252137, 7.71342670613578346275128236124, 8.406775288103347904102964578023, 9.146795587603413720654597412551