L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−2.54 + 0.717i)7-s − 0.999·8-s + (−5.09 − 2.94i)11-s + 4.05·13-s + (−1.89 − 1.84i)14-s + (−0.5 − 0.866i)16-s + (−0.371 − 0.214i)17-s + (−5.30 + 3.06i)19-s − 5.88i·22-s + (0.876 + 1.51i)23-s + (2.02 + 3.51i)26-s + (0.651 − 2.56i)28-s + 0.0419i·29-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.962 + 0.271i)7-s − 0.353·8-s + (−1.53 − 0.886i)11-s + 1.12·13-s + (−0.506 − 0.493i)14-s + (−0.125 − 0.216i)16-s + (−0.0901 − 0.0520i)17-s + (−1.21 + 0.703i)19-s − 1.25i·22-s + (0.182 + 0.316i)23-s + (0.397 + 0.689i)26-s + (0.123 − 0.484i)28-s + 0.00778i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.296741980\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.296741980\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.54 - 0.717i)T \) |
good | 11 | \( 1 + (5.09 + 2.94i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4.05T + 13T^{2} \) |
| 17 | \( 1 + (0.371 + 0.214i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.30 - 3.06i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.876 - 1.51i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 0.0419iT - 29T^{2} \) |
| 31 | \( 1 + (-7.92 - 4.57i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.928 - 0.536i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8.61T + 41T^{2} \) |
| 43 | \( 1 + 11.0iT - 43T^{2} \) |
| 47 | \( 1 + (-0.834 + 0.481i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.57 + 11.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.77 + 11.7i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.05 + 0.609i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.9 + 6.32i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.54iT - 71T^{2} \) |
| 73 | \( 1 + (-4.66 + 8.08i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.35 - 9.28i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 + (3.15 + 5.46i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2.59T + 97T^{2} \) |
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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
−8.436308406439179641438432193232, −8.088197937786961028505424315676, −7.00193551670167206136011866348, −6.27868009570827519500149593279, −5.76646916197183777385417733160, −5.01968802551127059607314140918, −3.85970470192378654486212007881, −3.23912962915868544016193281607, −2.28821911513956848597004415782, −0.44451454640367227495175950257,
0.901497623208167433432595597644, 2.44518297567055554712447998822, 2.83939938883491637181975728627, 4.13822229285877941384120349627, 4.52654110871430658038560005374, 5.71020719134966144993369251148, 6.28368346002039486472286499184, 7.14016618603056949263483501638, 8.004385336213393004200425883964, 8.816757437735764745179441127631