| L(s) = 1 | + 3-s + 2·5-s + 9-s + 2·11-s + 2·15-s − 6·17-s − 8·19-s + 6·23-s − 25-s + 27-s − 10·29-s − 4·31-s + 2·33-s − 6·37-s − 6·41-s − 4·43-s + 2·45-s − 8·47-s − 6·51-s + 2·53-s + 4·55-s − 8·57-s + 4·59-s + 8·61-s − 8·67-s + 6·69-s − 10·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.603·11-s + 0.516·15-s − 1.45·17-s − 1.83·19-s + 1.25·23-s − 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.718·31-s + 0.348·33-s − 0.986·37-s − 0.937·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s − 0.840·51-s + 0.274·53-s + 0.539·55-s − 1.05·57-s + 0.520·59-s + 1.02·61-s − 0.977·67-s + 0.722·69-s − 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.031650813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.031650813\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−12.76703744736846, −11.94950102357922, −11.43260912365849, −11.08602544352900, −10.62049672760470, −10.13292018130203, −9.705369391010510, −9.125872364770501, −8.856370561546684, −8.590675954927272, −8.003364082212139, −7.241967959065734, −6.872673222899412, −6.592002975682428, −6.020812405044393, −5.443779980531621, −5.029159081584965, −4.281175935442844, −4.043362565917578, −3.385172792336319, −2.817136328341067, −2.087519087178546, −1.852962389754267, −1.425080039732096, −0.2208575414547275,
0.2208575414547275, 1.425080039732096, 1.852962389754267, 2.087519087178546, 2.817136328341067, 3.385172792336319, 4.043362565917578, 4.281175935442844, 5.029159081584965, 5.443779980531621, 6.020812405044393, 6.592002975682428, 6.872673222899412, 7.241967959065734, 8.003364082212139, 8.590675954927272, 8.856370561546684, 9.125872364770501, 9.705369391010510, 10.13292018130203, 10.62049672760470, 11.08602544352900, 11.43260912365849, 11.94950102357922, 12.76703744736846
