L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.5 + 0.866i)3-s − 1.00i·4-s + (0.258 − 0.965i)5-s + (−0.258 − 0.965i)6-s + (1.22 + 1.22i)7-s + (0.707 + 0.707i)8-s + (−0.499 − 0.866i)9-s + (0.500 + 0.866i)10-s + (0.866 + 0.500i)12-s + (−0.707 + 0.707i)13-s − 1.73·14-s + (0.707 + 0.707i)15-s − 1.00·16-s + (1.36 − 1.36i)17-s + (0.965 + 0.258i)18-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.5 + 0.866i)3-s − 1.00i·4-s + (0.258 − 0.965i)5-s + (−0.258 − 0.965i)6-s + (1.22 + 1.22i)7-s + (0.707 + 0.707i)8-s + (−0.499 − 0.866i)9-s + (0.500 + 0.866i)10-s + (0.866 + 0.500i)12-s + (−0.707 + 0.707i)13-s − 1.73·14-s + (0.707 + 0.707i)15-s − 1.00·16-s + (1.36 − 1.36i)17-s + (0.965 + 0.258i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7860611808\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7860611808\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.258 + 0.965i)T \) |
| 13 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-0.366 + 0.366i)T - iT^{2} \) |
| 47 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 0.517iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.533747500648098764920016153603, −9.144014643243389240902360955699, −8.308905334667721765803186974654, −7.69113210807765209085677348665, −6.37478763452037745790038123464, −5.57477298950555973417428951185, −4.93933825968215458540336984040, −4.59742751070505559852995880860, −2.58912041684515195329583853940, −1.22833446216149373939247225977,
1.08146948214631769090700715533, 2.02654155679185455709451600226, 3.14551158622809556762196079911, 4.25131091571927345186359130591, 5.40638952302217120315135613600, 6.50764723084254360384974548339, 7.34459121069616355709142847360, 7.87402041183533113991544040891, 8.285174454922438708205088821065, 9.936509640573880412490338668949