L(s) = 1 | + 3.52·5-s + (1.16 + 2.37i)7-s + 2.32·11-s + (−2.35 − 4.08i)13-s + (0.636 + 1.10i)17-s + (2.78 − 4.82i)19-s + 3.29·23-s + 7.45·25-s + (4.32 − 7.48i)29-s + (−4.25 + 7.37i)31-s + (4.11 + 8.38i)35-s + (−2.84 + 4.91i)37-s + (−1.66 − 2.88i)41-s + (0.0444 − 0.0769i)43-s + (3.52 + 6.10i)47-s + ⋯ |
L(s) = 1 | + 1.57·5-s + (0.441 + 0.897i)7-s + 0.699·11-s + (−0.654 − 1.13i)13-s + (0.154 + 0.267i)17-s + (0.638 − 1.10i)19-s + 0.687·23-s + 1.49·25-s + (0.802 − 1.38i)29-s + (−0.764 + 1.32i)31-s + (0.696 + 1.41i)35-s + (−0.466 + 0.808i)37-s + (−0.260 − 0.450i)41-s + (0.00677 − 0.0117i)43-s + (0.514 + 0.890i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.505727635\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.505727635\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.16 - 2.37i)T \) |
good | 5 | \( 1 - 3.52T + 5T^{2} \) |
| 11 | \( 1 - 2.32T + 11T^{2} \) |
| 13 | \( 1 + (2.35 + 4.08i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.636 - 1.10i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.78 + 4.82i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 3.29T + 23T^{2} \) |
| 29 | \( 1 + (-4.32 + 7.48i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.25 - 7.37i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.84 - 4.91i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.66 + 2.88i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.0444 + 0.0769i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.52 - 6.10i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.41 + 5.92i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.99 - 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.67 + 11.5i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.06 - 5.30i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 1.30T + 71T^{2} \) |
| 73 | \( 1 + (-6.64 - 11.5i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.01 - 8.68i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.90 + 10.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.561 - 0.972i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.50 - 6.07i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.412079725345601429769341497578, −8.918292526582738234861690195482, −7.992641513574706866967293807802, −6.89533485016084737427332959033, −6.12524384154149408813131962110, −5.33041557444654715814454242822, −4.84110862332050664041532508020, −3.11457499565901127129046243009, −2.36290452416616982150906369192, −1.25236861432813743109547232837,
1.30669829559258387089132237429, 2.04559900240168328513458737872, 3.42718602808413399025714483896, 4.53809947467102750428862643912, 5.32794302894566850130363361990, 6.24755141163019951908843646275, 6.98114173680744758764847814299, 7.70296150280021275971311835143, 9.055091085116038684317106470515, 9.359879626900771730653034511495