L(s) = 1 | − 4·7-s − 4·11-s + 4·13-s + 6·17-s − 4·19-s + 4·23-s − 4·29-s + 4·37-s + 8·41-s + 12·47-s + 9·49-s + 2·53-s + 12·59-s + 2·61-s + 8·67-s + 8·71-s − 16·73-s + 16·77-s − 8·79-s + 8·83-s − 16·91-s − 8·97-s − 12·101-s − 4·103-s + 8·107-s + 18·109-s + 10·113-s + ⋯ |
L(s) = 1 | − 1.51·7-s − 1.20·11-s + 1.10·13-s + 1.45·17-s − 0.917·19-s + 0.834·23-s − 0.742·29-s + 0.657·37-s + 1.24·41-s + 1.75·47-s + 9/7·49-s + 0.274·53-s + 1.56·59-s + 0.256·61-s + 0.977·67-s + 0.949·71-s − 1.87·73-s + 1.82·77-s − 0.900·79-s + 0.878·83-s − 1.67·91-s − 0.812·97-s − 1.19·101-s − 0.394·103-s + 0.773·107-s + 1.72·109-s + 0.940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.308631451\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.308631451\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.314029715617313776311627017378, −8.540069133879418786215322313611, −7.66632465262567351302446068232, −6.89816910922858703609249536824, −5.93716469340883286987149149195, −5.51156754827561892754115225807, −4.11413006742628847488506548763, −3.30901127102544303971090199635, −2.48554463513076455727425590618, −0.76691162216153136756395979460,
0.76691162216153136756395979460, 2.48554463513076455727425590618, 3.30901127102544303971090199635, 4.11413006742628847488506548763, 5.51156754827561892754115225807, 5.93716469340883286987149149195, 6.89816910922858703609249536824, 7.66632465262567351302446068232, 8.540069133879418786215322313611, 9.314029715617313776311627017378