L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.668 + 0.668i)3-s − 1.00i·4-s + (−2.17 − 0.508i)5-s + 0.945i·6-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + 2.10i·9-s + (−1.89 + 1.17i)10-s + (2.68 − 1.94i)11-s + (0.668 + 0.668i)12-s + (1.94 + 1.94i)13-s + 1.00i·14-s + (1.79 − 1.11i)15-s − 1.00·16-s + (3.61 − 3.61i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.386 + 0.386i)3-s − 0.500i·4-s + (−0.973 − 0.227i)5-s + 0.386i·6-s + (−0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.701i·9-s + (−0.600 + 0.373i)10-s + (0.809 − 0.586i)11-s + (0.193 + 0.193i)12-s + (0.539 + 0.539i)13-s + 0.267i·14-s + (0.463 − 0.288i)15-s − 0.250·16-s + (0.875 − 0.875i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.47179 - 0.471456i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47179 - 0.471456i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.17 + 0.508i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
| 11 | \( 1 + (-2.68 + 1.94i)T \) |
good | 3 | \( 1 + (0.668 - 0.668i)T - 3iT^{2} \) |
| 13 | \( 1 + (-1.94 - 1.94i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.61 + 3.61i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.79T + 19T^{2} \) |
| 23 | \( 1 + (-6.28 + 6.28i)T - 23iT^{2} \) |
| 29 | \( 1 - 4.14T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 + (6.35 + 6.35i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.80iT - 41T^{2} \) |
| 43 | \( 1 + (-8.15 - 8.15i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.337 + 0.337i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.85 + 5.85i)T - 53iT^{2} \) |
| 59 | \( 1 - 13.4iT - 59T^{2} \) |
| 61 | \( 1 + 0.995iT - 61T^{2} \) |
| 67 | \( 1 + (2.69 + 2.69i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.97T + 71T^{2} \) |
| 73 | \( 1 + (3.48 + 3.48i)T + 73iT^{2} \) |
| 79 | \( 1 - 8.68T + 79T^{2} \) |
| 83 | \( 1 + (-8.86 - 8.86i)T + 83iT^{2} \) |
| 89 | \( 1 + 6.56iT - 89T^{2} \) |
| 97 | \( 1 + (5.10 + 5.10i)T + 97iT^{2} \) |
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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
−10.56934510175579039798841535281, −9.383916174813688607436218511978, −8.747544895690562266942709216620, −7.60572666508326490927874399262, −6.63086304505735771950413127440, −5.49436650251305330570497191693, −4.73263797053508916791090762594, −3.79874673630529532898388085671, −2.84697849604628933416068066010, −0.967983795942120570987465177448,
1.10334643480776330252216520403, 3.42225067501817571966675454875, 3.72804018725637633893927528722, 5.14848095985685918246567442802, 6.07408889243850105835889403309, 7.06630178280751591847659616752, 7.39096646577532464220216436877, 8.535436902300272935665021970349, 9.426023976121355489357976648161, 10.58478380387359254931908952354