L(s) = 1 | + (−1.53 + 1.53i)3-s + (−0.569 + 2.16i)5-s − 4.20·7-s − 1.73i·9-s + (4.38 − 4.38i)11-s + (2.17 − 2.17i)13-s + (−2.44 − 4.20i)15-s − 2.75i·17-s + (−1.93 − 1.93i)19-s + (6.46 − 6.46i)21-s − 3.37·23-s + (−4.35 − 2.46i)25-s + (−1.95 − 1.95i)27-s + (4.46 + 4.46i)29-s + 1.03·31-s + ⋯ |
L(s) = 1 | + (−0.888 + 0.888i)3-s + (−0.254 + 0.966i)5-s − 1.58·7-s − 0.577i·9-s + (1.32 − 1.32i)11-s + (0.603 − 0.603i)13-s + (−0.632 − 1.08i)15-s − 0.668i·17-s + (−0.443 − 0.443i)19-s + (1.41 − 1.41i)21-s − 0.704·23-s + (−0.870 − 0.492i)25-s + (−0.375 − 0.375i)27-s + (0.828 + 0.828i)29-s + 0.185·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7960372449\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7960372449\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.569 - 2.16i)T \) |
good | 3 | \( 1 + (1.53 - 1.53i)T - 3iT^{2} \) |
| 7 | \( 1 + 4.20T + 7T^{2} \) |
| 11 | \( 1 + (-4.38 + 4.38i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2.17 + 2.17i)T - 13iT^{2} \) |
| 17 | \( 1 + 2.75iT - 17T^{2} \) |
| 19 | \( 1 + (1.93 + 1.93i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.37T + 23T^{2} \) |
| 29 | \( 1 + (-4.46 - 4.46i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.03T + 31T^{2} \) |
| 37 | \( 1 + (-7.53 - 7.53i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.26iT - 41T^{2} \) |
| 43 | \( 1 + (-4.91 - 4.91i)T + 43iT^{2} \) |
| 47 | \( 1 + 0.301iT - 47T^{2} \) |
| 53 | \( 1 + (4.93 + 4.93i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.76 + 4.76i)T - 59iT^{2} \) |
| 61 | \( 1 + (2 + 2i)T + 61iT^{2} \) |
| 67 | \( 1 + (4.31 - 4.31i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.58iT - 71T^{2} \) |
| 73 | \( 1 - 4.77T + 73T^{2} \) |
| 79 | \( 1 + 3.86T + 79T^{2} \) |
| 83 | \( 1 + (-3.48 + 3.48i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.46iT - 89T^{2} \) |
| 97 | \( 1 + 15.8iT - 97T^{2} \) |
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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
−9.866195627340502866609121158841, −9.144911388593923246971597059555, −8.138193576337505276011091278227, −6.79806122961330903051544427202, −6.29181739937522701249243492936, −5.79437812889038442989091477441, −4.40730078163340312702245239151, −3.51177074392895729039496778739, −2.94935948542847078771790029637, −0.51927937875501291540270822466,
0.902453075224477609667881819638, 1.99346024736612021392321323270, 3.84581645563392772352836362441, 4.34548037527158071661660844897, 5.91064686013362610416758192261, 6.25919513117778916657373420284, 6.96025246558953652595203398905, 7.87397111758551560142378413978, 9.065460179725407081602058766092, 9.480990276941809550904137129356