L(s) = 1 | + (−0.366 + 1.36i)2-s + (−1.73 − i)4-s + (−1.86 − 1.23i)5-s + (2 − 1.99i)8-s + (−2.59 + 1.5i)9-s + (2.36 − 2.09i)10-s + (1 − i)13-s + (1.99 + 3.46i)16-s + (4.09 − 1.09i)17-s + (−1.09 − 4.09i)18-s + (2 + 4i)20-s + (1.96 + 4.59i)25-s + (1 + 1.73i)26-s + 4i·29-s + (−5.46 + 1.46i)32-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.866 − 0.5i)4-s + (−0.834 − 0.550i)5-s + (0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.748 − 0.663i)10-s + (0.277 − 0.277i)13-s + (0.499 + 0.866i)16-s + (0.993 − 0.266i)17-s + (−0.258 − 0.965i)18-s + (0.447 + 0.894i)20-s + (0.392 + 0.919i)25-s + (0.196 + 0.339i)26-s + 0.742i·29-s + (−0.965 + 0.258i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.517954 + 0.660073i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.517954 + 0.660073i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 5 | \( 1 + (1.86 + 1.23i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.09 + 1.09i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.56 - 9.56i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.29 - 12.2i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-4.02 - 15.0i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + (-13.8 + 8i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (13 + 13i)T + 97iT^{2} \) |
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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
−10.09768436472000509853174854508, −9.071034376482838217491851202417, −8.435666294431672199598254200965, −7.81663762264873537965298712483, −7.06803307519818886681158591176, −5.84560053950339525983148681579, −5.21702369564741684539490114235, −4.27463355152783802809991195544, −3.11798097105132635371657819136, −1.04080545308933562506111898808,
0.54888306258959967741179582848, 2.30196820499498107034285342555, 3.41026813961926222205522704825, 3.95104313842934157132971414662, 5.25732030553520661090493643061, 6.35294445817611710197647100692, 7.55452933186477464361343644250, 8.190034773308068240033329344737, 9.026774763357027835535211586563, 9.833059421083350282518883607420