L(s) = 1 | + (−0.366 − 0.366i)2-s − 0.732i·4-s + (−0.707 + 0.707i)7-s + (−0.633 + 0.633i)8-s + 1.93i·11-s + 0.517·14-s − 0.267·16-s + (0.707 − 0.707i)22-s + (−1.36 + 1.36i)23-s + (0.517 + 0.517i)28-s − 0.517·29-s + (0.732 + 0.732i)32-s + (−0.707 + 0.707i)37-s + (0.707 + 0.707i)43-s + 1.41·44-s + ⋯ |
L(s) = 1 | + (−0.366 − 0.366i)2-s − 0.732i·4-s + (−0.707 + 0.707i)7-s + (−0.633 + 0.633i)8-s + 1.93i·11-s + 0.517·14-s − 0.267·16-s + (0.707 − 0.707i)22-s + (−1.36 + 1.36i)23-s + (0.517 + 0.517i)28-s − 0.517·29-s + (0.732 + 0.732i)32-s + (−0.707 + 0.707i)37-s + (0.707 + 0.707i)43-s + 1.41·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5824026354\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5824026354\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 2 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 11 | \( 1 - 1.93iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
| 29 | \( 1 + 0.517T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 71 | \( 1 - 0.517iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 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
−9.746390656935353388900641372813, −9.346142775819755846682483830049, −8.329383115485110878137850860668, −7.32239974841637978429794039386, −6.50927997815118777455963151415, −5.65399203099012365497566509602, −4.92206129013777442742288408623, −3.75979719155097544897492757450, −2.41625133245544122439162071957, −1.71537143237607301999522158749,
0.50012489465889861143531675321, 2.61439408513177880902926591006, 3.57696924572056915400755650890, 4.12527080358336090050038417874, 5.70693674536575711845824443710, 6.34083352123632998494843505776, 7.13714814090795498483808518038, 7.967846832131971277377909073181, 8.622751879436082587289266151492, 9.227919670187391511678118573687