L(s) = 1 | + 5-s − 7-s − 6·13-s + 2·17-s + 4·19-s − 23-s + 25-s + 2·29-s + 8·31-s − 35-s + 10·37-s − 2·41-s + 4·43-s − 4·47-s + 49-s + 6·53-s + 4·59-s + 10·61-s − 6·65-s + 12·67-s − 8·71-s − 2·73-s − 12·79-s + 4·83-s + 2·85-s + 18·89-s + 6·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 1.66·13-s + 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.169·35-s + 1.64·37-s − 0.312·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s + 1.28·61-s − 0.744·65-s + 1.46·67-s − 0.949·71-s − 0.234·73-s − 1.35·79-s + 0.439·83-s + 0.216·85-s + 1.90·89-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.760845887\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.760845887\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.50511446697820, −13.13896395037142, −12.70188244042231, −11.98535761864770, −11.85644782969241, −11.31508256611095, −10.46854909020984, −10.05044661768033, −9.778614288134479, −9.393423753801296, −8.695108305211412, −8.136374647238496, −7.532699622500643, −7.241876750163854, −6.534363965382831, −6.103048424451263, −5.469332634908085, −4.969856821870748, −4.522135688596509, −3.788327599494274, −3.073880693783199, −2.558891715999024, −2.139904633962308, −1.091100642492584, −0.5730423300549051,
0.5730423300549051, 1.091100642492584, 2.139904633962308, 2.558891715999024, 3.073880693783199, 3.788327599494274, 4.522135688596509, 4.969856821870748, 5.469332634908085, 6.103048424451263, 6.534363965382831, 7.241876750163854, 7.532699622500643, 8.136374647238496, 8.695108305211412, 9.393423753801296, 9.778614288134479, 10.05044661768033, 10.46854909020984, 11.31508256611095, 11.85644782969241, 11.98535761864770, 12.70188244042231, 13.13896395037142, 13.50511446697820