L(s) = 1 | + (0.834 − 0.223i)3-s + (2.20 − 0.382i)5-s + (2.07 − 1.19i)7-s + (−1.95 + 1.12i)9-s + (0.0379 + 0.141i)11-s + (−3.58 + 0.344i)13-s + (1.75 − 0.811i)15-s + (1.33 − 4.96i)17-s + (4.18 + 1.12i)19-s + (1.46 − 1.46i)21-s + (2.28 + 8.53i)23-s + (4.70 − 1.68i)25-s + (−3.20 + 3.20i)27-s + (−5.00 − 2.88i)29-s + (−4.94 − 4.94i)31-s + ⋯ |
L(s) = 1 | + (0.481 − 0.129i)3-s + (0.985 − 0.171i)5-s + (0.783 − 0.452i)7-s + (−0.650 + 0.375i)9-s + (0.0114 + 0.0427i)11-s + (−0.995 + 0.0956i)13-s + (0.452 − 0.209i)15-s + (0.322 − 1.20i)17-s + (0.960 + 0.257i)19-s + (0.318 − 0.318i)21-s + (0.476 + 1.77i)23-s + (0.941 − 0.337i)25-s + (−0.617 + 0.617i)27-s + (−0.928 − 0.536i)29-s + (−0.887 − 0.887i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.65901 - 0.216875i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.65901 - 0.216875i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.20 + 0.382i)T \) |
| 13 | \( 1 + (3.58 - 0.344i)T \) |
good | 3 | \( 1 + (-0.834 + 0.223i)T + (2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (-2.07 + 1.19i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0379 - 0.141i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.33 + 4.96i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.18 - 1.12i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.28 - 8.53i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (5.00 + 2.88i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.94 + 4.94i)T + 31iT^{2} \) |
| 37 | \( 1 + (-3.76 - 2.17i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (11.6 - 3.11i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.642 - 0.172i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 - 3.88iT - 47T^{2} \) |
| 53 | \( 1 + (2.38 + 2.38i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.94 - 7.26i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (5.54 + 9.61i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.92 - 3.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.66 + 6.21i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 - 0.839T + 73T^{2} \) |
| 79 | \( 1 + 3.64iT - 79T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 + (11.7 - 3.15i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (0.254 + 0.440i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.79543344232184062846697388886, −11.12262686452780623320502702922, −9.717600318725400754508813621719, −9.328970851904908789933937200543, −7.893212286791023322509579761117, −7.29542398296079552907373077486, −5.62263108587678815958422664357, −4.92644184011180749827072350589, −3.07787648255529334789417087150, −1.74092905913134573681512071315,
1.98335022780897355120309935801, 3.19333844098892573125057195413, 4.96391913670128915214920195195, 5.83248289192419170343278344293, 7.08766911659156786977509287736, 8.402490847729018072918146213671, 9.052390961733861302335249311302, 10.07125472794795383000649218430, 10.97501562240581417121484647541, 12.10228358431490362711725592629