L(s) = 1 | + (−0.183 + 0.183i)3-s + 3.84i·7-s + 2.93i·9-s + (−1.60 − 1.60i)11-s + (1.80 − 1.80i)13-s + 4.93·17-s + (−4.77 + 4.77i)19-s + (−0.707 − 0.707i)21-s − 0.134i·23-s + (−1.09 − 1.09i)27-s + (−2.17 + 2.17i)29-s − 2.26·31-s + 0.588·33-s + (−4.35 − 4.35i)37-s + 0.664i·39-s + ⋯ |
L(s) = 1 | + (−0.106 + 0.106i)3-s + 1.45i·7-s + 0.977i·9-s + (−0.482 − 0.482i)11-s + (0.501 − 0.501i)13-s + 1.19·17-s + (−1.09 + 1.09i)19-s + (−0.154 − 0.154i)21-s − 0.0280i·23-s + (−0.209 − 0.209i)27-s + (−0.403 + 0.403i)29-s − 0.406·31-s + 0.102·33-s + (−0.715 − 0.715i)37-s + 0.106i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.082342236\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.082342236\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.183 - 0.183i)T - 3iT^{2} \) |
| 7 | \( 1 - 3.84iT - 7T^{2} \) |
| 11 | \( 1 + (1.60 + 1.60i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.80 + 1.80i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.93T + 17T^{2} \) |
| 19 | \( 1 + (4.77 - 4.77i)T - 19iT^{2} \) |
| 23 | \( 1 + 0.134iT - 23T^{2} \) |
| 29 | \( 1 + (2.17 - 2.17i)T - 29iT^{2} \) |
| 31 | \( 1 + 2.26T + 31T^{2} \) |
| 37 | \( 1 + (4.35 + 4.35i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.34iT - 41T^{2} \) |
| 43 | \( 1 + (-2.70 - 2.70i)T + 43iT^{2} \) |
| 47 | \( 1 + 7.03T + 47T^{2} \) |
| 53 | \( 1 + (3.40 + 3.40i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.107 + 0.107i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.46 - 3.46i)T - 61iT^{2} \) |
| 67 | \( 1 + (-1.91 + 1.91i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.32iT - 71T^{2} \) |
| 73 | \( 1 - 9.82iT - 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + (-8.80 + 8.80i)T - 83iT^{2} \) |
| 89 | \( 1 + 1.12iT - 89T^{2} \) |
| 97 | \( 1 + 6.10T + 97T^{2} \) |
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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
−9.773278202334389392782161129066, −8.710962362962154756126898707041, −8.244793513331765298682173900762, −7.55100053271948578587046874306, −6.16116967433239793680877341920, −5.63476526659320350004501859014, −5.02561476304388160912839694024, −3.67745750125413632367679210622, −2.70956461859220300436811459789, −1.70585808061453445286504668740,
0.41884583728009316824912946815, 1.65097475132529507388118406639, 3.20482432092340129337173874822, 4.00712609587173946299241985795, 4.79988423598526255200371774448, 5.98849031964415597832231636375, 6.82297447146460503704566647698, 7.34039362860626734634787028436, 8.247293264037537570893146657108, 9.201837200895472199505662905079