L(s) = 1 | − 5·7-s + 2·11-s + 13-s − 3·17-s − 2·19-s + 4·23-s + 6·29-s − 4·31-s − 11·37-s − 8·41-s + 43-s + 9·47-s + 18·49-s − 12·53-s − 6·59-s − 6·67-s − 7·71-s + 2·73-s − 10·77-s + 12·79-s − 16·83-s + 10·89-s − 5·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 1.88·7-s + 0.603·11-s + 0.277·13-s − 0.727·17-s − 0.458·19-s + 0.834·23-s + 1.11·29-s − 0.718·31-s − 1.80·37-s − 1.24·41-s + 0.152·43-s + 1.31·47-s + 18/7·49-s − 1.64·53-s − 0.781·59-s − 0.733·67-s − 0.830·71-s + 0.234·73-s − 1.13·77-s + 1.35·79-s − 1.75·83-s + 1.05·89-s − 0.524·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9575428888\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9575428888\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.50295880058460, −15.15226094579610, −14.21892855322139, −13.80941085850588, −13.26157132716824, −12.74154145199710, −12.32893365174152, −11.78488036723741, −10.98488102005989, −10.42678803226842, −10.05919736939236, −9.144326638576527, −9.049446308312792, −8.438597147926197, −7.367879912216263, −6.933412727649895, −6.372469110671409, −6.050056263428961, −5.114320054582475, −4.415305787615169, −3.570852710432322, −3.254272345442026, −2.443956269543023, −1.491021712038275, −0.3910713579752760,
0.3910713579752760, 1.491021712038275, 2.443956269543023, 3.254272345442026, 3.570852710432322, 4.415305787615169, 5.114320054582475, 6.050056263428961, 6.372469110671409, 6.933412727649895, 7.367879912216263, 8.438597147926197, 9.049446308312792, 9.144326638576527, 10.05919736939236, 10.42678803226842, 10.98488102005989, 11.78488036723741, 12.32893365174152, 12.74154145199710, 13.26157132716824, 13.80941085850588, 14.21892855322139, 15.15226094579610, 15.50295880058460