L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)8-s − 1.00·16-s − 1.41·17-s + (−1 − i)19-s − 1.41·23-s − 2i·31-s + (0.707 − 0.707i)32-s + (1.00 − 1.00i)34-s + 1.41·38-s + (1.00 − 1.00i)46-s + 1.41i·47-s − 49-s + (−1 + i)61-s + (1.41 + 1.41i)62-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)8-s − 1.00·16-s − 1.41·17-s + (−1 − i)19-s − 1.41·23-s − 2i·31-s + (0.707 − 0.707i)32-s + (1.00 − 1.00i)34-s + 1.41·38-s + (1.00 − 1.00i)46-s + 1.41i·47-s − 49-s + (−1 + i)61-s + (1.41 + 1.41i)62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2434917459\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2434917459\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 + (1 + i)T + iT^{2} \) |
| 23 | \( 1 + 1.41T + T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + 2iT - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (1 - i)T - iT^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−8.479955180473945963497386086152, −7.81080650170841406206213795148, −7.07475716280327681707089289740, −6.26604891605964461487472289660, −5.87879829589252799161627287048, −4.61350983078824726940189166123, −4.24353879066118819682560675171, −2.60723553073464479934983299064, −1.81600961662093426441045057405, −0.16671518267732937658998879965,
1.61437694517079147150296135734, 2.30813523539259161066549742251, 3.44094805437287590450927038824, 4.14603874487716110791227026129, 4.98217247665878702418684643061, 6.24597591718127930628764576941, 6.78853433220247096546344185676, 7.71909858512125339816937232839, 8.476538488618976638541710143482, 8.807908588071774236629396179495