L(s) = 1 | + (0.707 + 0.707i)2-s + (0.923 − 0.382i)3-s + 1.00i·4-s + (−0.382 + 0.923i)5-s + (0.923 + 0.382i)6-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)10-s + (0.382 + 0.923i)12-s + (−1.30 − 1.30i)13-s + 1.00·14-s + i·15-s − 1.00·16-s + 18-s + 0.765i·19-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (0.923 − 0.382i)3-s + 1.00i·4-s + (−0.382 + 0.923i)5-s + (0.923 + 0.382i)6-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)10-s + (0.382 + 0.923i)12-s + (−1.30 − 1.30i)13-s + 1.00·14-s + i·15-s − 1.00·16-s + 18-s + 0.765i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.742212542\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.742212542\) |
\(L(1)\) |
|
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 | \( 1 + (-0.923 + 0.382i)T \) |
| 5 | \( 1 + (0.382 - 0.923i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - 0.765iT - T^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + 0.765T + T^{2} \) |
| 61 | \( 1 - 1.84T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 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
−10.45362254082624704172055619897, −9.718704304859890964433570794525, −8.319287410440639393894778689005, −7.66667762972638475235669069658, −7.44266842185411688767053165566, −6.39206330035907945201523246089, −5.23519761437822161147227665876, −4.04660642380497022441242921538, −3.32385379317349383642297516461, −2.25480452861876754360783634744,
1.83630514535151178492272383011, 2.60937578318371307073940951499, 4.10831182486204809594423820310, 4.61880350897540286663537687366, 5.35672777044677713320924898954, 6.81963766864865702319758876895, 7.952193204480424376914806341819, 8.863790145333007437534645384449, 9.361527105159831421316789219682, 10.18884991007780340949458464042