L(s) = 1 | + (−0.118 − 0.673i)3-s + (0.500 − 0.181i)9-s + (−0.300 + 0.173i)11-s + (0.939 + 0.342i)17-s + (0.642 − 0.766i)19-s + (0.173 − 0.984i)25-s + (−0.524 − 0.907i)27-s + (0.152 + 0.181i)33-s + (−1.26 + 0.223i)41-s + (−0.642 − 0.766i)43-s + (0.5 + 0.866i)49-s + (0.118 − 0.673i)51-s + (−0.592 − 0.342i)57-s + (1.85 + 0.673i)59-s + (−1.20 + 0.439i)67-s + ⋯ |
L(s) = 1 | + (−0.118 − 0.673i)3-s + (0.500 − 0.181i)9-s + (−0.300 + 0.173i)11-s + (0.939 + 0.342i)17-s + (0.642 − 0.766i)19-s + (0.173 − 0.984i)25-s + (−0.524 − 0.907i)27-s + (0.152 + 0.181i)33-s + (−1.26 + 0.223i)41-s + (−0.642 − 0.766i)43-s + (0.5 + 0.866i)49-s + (0.118 − 0.673i)51-s + (−0.592 − 0.342i)57-s + (1.85 + 0.673i)59-s + (−1.20 + 0.439i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.075170819\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.075170819\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-0.642 + 0.766i)T \) |
good | 3 | \( 1 + (0.118 + 0.673i)T + (-0.939 + 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.300 - 0.173i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (1.26 - 0.223i)T + (0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (0.642 + 0.766i)T + (-0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (-1.85 - 0.673i)T + (0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (1.20 - 0.439i)T + (0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (-1.32 - 0.766i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (1.70 + 0.300i)T + (0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.439 + 1.20i)T + (-0.766 - 0.642i)T^{2} \) |
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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
−9.969514042149521263956557784255, −8.939611550761479849556239271869, −8.046502811114920795933334280571, −7.30162887246990802591565473317, −6.63545079241196165008517644445, −5.68345514636668662147246012302, −4.73127912622031973335850489679, −3.62385993646434167842761526153, −2.42113612990466492892477293578, −1.13939779746055201226335974365,
1.55430523034076738506449401481, 3.13037903922762753027582431588, 3.91317771474310893001751360321, 5.07669295973534787727807711991, 5.54754355404787220953416898462, 6.83027372430058272086083227719, 7.61002286221607804535059589091, 8.423591023220750115404716779051, 9.500412228861090241070300055302, 9.985790194545231919556344350360