L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 + 0.707i)8-s + (−1.22 + 0.707i)11-s + (0.500 − 0.866i)16-s + (0.999 − i)22-s + (0.707 − 1.22i)23-s + (−0.5 − 0.866i)25-s + 1.41·29-s + (−0.258 + 0.965i)32-s + (1.73 + i)37-s + (−0.707 + 1.22i)44-s + (−0.366 + 1.36i)46-s + (0.707 + 0.707i)50-s + (0.707 + 1.22i)53-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 + 0.707i)8-s + (−1.22 + 0.707i)11-s + (0.500 − 0.866i)16-s + (0.999 − i)22-s + (0.707 − 1.22i)23-s + (−0.5 − 0.866i)25-s + 1.41·29-s + (−0.258 + 0.965i)32-s + (1.73 + i)37-s + (−0.707 + 1.22i)44-s + (−0.366 + 1.36i)46-s + (0.707 + 0.707i)50-s + (0.707 + 1.22i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7537335739\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7537335739\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.576894926304923065699275308817, −8.186200661286028410879952625935, −7.44489035779570947389239096878, −6.70408432241950939820858426182, −6.03879332750755548456038818563, −5.08316626079114122221429638644, −4.37616833819932999060556296057, −2.79066412955536811545513521573, −2.37038297846789174328206373223, −0.908463878619858256183076698821,
0.823239394549130815403278841678, 2.12752316644489962314444327526, 2.98589715333756599241069348626, 3.73620790976095700264468687548, 5.05452067541768032546500491939, 5.79133545683087927219618318688, 6.63677961318306715547023793356, 7.52847520258263196176481585529, 7.970736788097982290236420100615, 8.682381917468510753125745933729