L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s + 13-s − 15-s − 6·17-s + 8·19-s − 4·23-s + 25-s − 27-s − 2·29-s + 4·31-s + 4·33-s + 10·37-s − 39-s − 6·41-s − 8·43-s + 45-s + 8·47-s + 6·51-s − 2·53-s − 4·55-s − 8·57-s − 8·59-s + 2·61-s + 65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.258·15-s − 1.45·17-s + 1.83·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.696·33-s + 1.64·37-s − 0.160·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s + 1.16·47-s + 0.840·51-s − 0.274·53-s − 0.539·55-s − 1.05·57-s − 1.04·59-s + 0.256·61-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.783724032\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.783724032\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.43393453075604, −12.95285369690332, −12.27678012011827, −11.93010766631379, −11.31884589445999, −11.02786370267336, −10.43781285441713, −10.04681829506308, −9.507233952462686, −9.173358714004592, −8.370930986273367, −7.993595564216868, −7.432928125116166, −6.980299629044200, −6.212902627022794, −6.042876326689095, −5.368553082359104, −4.792384313980657, −4.610817946604804, −3.606639553271793, −3.196618194581615, −2.306601009792963, −2.050804592021997, −1.061834327704170, −0.4529486176202810,
0.4529486176202810, 1.061834327704170, 2.050804592021997, 2.306601009792963, 3.196618194581615, 3.606639553271793, 4.610817946604804, 4.792384313980657, 5.368553082359104, 6.042876326689095, 6.212902627022794, 6.980299629044200, 7.432928125116166, 7.993595564216868, 8.370930986273367, 9.173358714004592, 9.507233952462686, 10.04681829506308, 10.43781285441713, 11.02786370267336, 11.31884589445999, 11.93010766631379, 12.27678012011827, 12.95285369690332, 13.43393453075604