L(s) = 1 | + 2-s + (−1.05 − 1.05i)3-s + 4-s + (−2.15 + 0.609i)5-s + (−1.05 − 1.05i)6-s + (−1.21 − 1.21i)7-s + 8-s − 0.789i·9-s + (−2.15 + 0.609i)10-s − 4.85i·11-s + (−1.05 − 1.05i)12-s − 5.57·13-s + (−1.21 − 1.21i)14-s + (2.90 + 1.62i)15-s + 16-s + 1.00i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.606 − 0.606i)3-s + 0.5·4-s + (−0.962 + 0.272i)5-s + (−0.429 − 0.429i)6-s + (−0.460 − 0.460i)7-s + 0.353·8-s − 0.263i·9-s + (−0.680 + 0.192i)10-s − 1.46i·11-s + (−0.303 − 0.303i)12-s − 1.54·13-s + (−0.325 − 0.325i)14-s + (0.749 + 0.418i)15-s + 0.250·16-s + 0.244i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.398917 - 0.870153i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.398917 - 0.870153i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + (2.15 - 0.609i)T \) |
| 37 | \( 1 + (-4.75 - 3.78i)T \) |
good | 3 | \( 1 + (1.05 + 1.05i)T + 3iT^{2} \) |
| 7 | \( 1 + (1.21 + 1.21i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.85iT - 11T^{2} \) |
| 13 | \( 1 + 5.57T + 13T^{2} \) |
| 17 | \( 1 - 1.00iT - 17T^{2} \) |
| 19 | \( 1 + (-1.94 + 1.94i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.14T + 23T^{2} \) |
| 29 | \( 1 + (4.77 + 4.77i)T + 29iT^{2} \) |
| 31 | \( 1 + (1.71 - 1.71i)T - 31iT^{2} \) |
| 41 | \( 1 + 0.412iT - 41T^{2} \) |
| 43 | \( 1 + 1.47T + 43T^{2} \) |
| 47 | \( 1 + (-8.35 - 8.35i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.85 + 3.85i)T - 53iT^{2} \) |
| 59 | \( 1 + (-7.33 + 7.33i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.457 - 0.457i)T - 61iT^{2} \) |
| 67 | \( 1 + (5.29 - 5.29i)T - 67iT^{2} \) |
| 71 | \( 1 - 7.15T + 71T^{2} \) |
| 73 | \( 1 + (9.08 + 9.08i)T + 73iT^{2} \) |
| 79 | \( 1 + (-1.46 + 1.46i)T - 79iT^{2} \) |
| 83 | \( 1 + (-9.88 + 9.88i)T - 83iT^{2} \) |
| 89 | \( 1 + (-7.09 - 7.09i)T + 89iT^{2} \) |
| 97 | \( 1 - 2.10iT - 97T^{2} \) |
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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
−11.44025454778102343317832987136, −10.53146211806279818247985639183, −9.249948502306241614088707133772, −7.83778221472676992914336345454, −7.10949172730758349357898633479, −6.31344929367717476601158211328, −5.22432166914915956110701093913, −3.90037076172018305211656057235, −2.91751639459033883558920369938, −0.53683566094974935167656753409,
2.45303344521766376839936733834, 3.97651588466535562592167207471, 4.85948881106752383841354456297, 5.48995795618856468318852091517, 7.12053526956289888278442634235, 7.59091624051954923005503005135, 9.206030841530704241542312736301, 10.03113604449607849471283015083, 10.97727339991396819208320836449, 11.98278889597603173452479789102