| L(s) = 1 | + 2·3-s + 2·5-s + 7-s + 9-s − 11-s + 4·15-s + 4·17-s + 4·19-s + 2·21-s − 4·23-s − 25-s − 4·27-s + 2·29-s − 2·31-s − 2·33-s + 2·35-s − 6·37-s + 4·41-s − 4·43-s + 2·45-s + 2·47-s + 49-s + 8·51-s + 2·53-s − 2·55-s + 8·57-s − 6·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.03·15-s + 0.970·17-s + 0.917·19-s + 0.436·21-s − 0.834·23-s − 1/5·25-s − 0.769·27-s + 0.371·29-s − 0.359·31-s − 0.348·33-s + 0.338·35-s − 0.986·37-s + 0.624·41-s − 0.609·43-s + 0.298·45-s + 0.291·47-s + 1/7·49-s + 1.12·51-s + 0.274·53-s − 0.269·55-s + 1.05·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.483500673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.483500673\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−10.28981135359909303761462495429, −9.721026942562876571917863787048, −8.900267729944967573194117141070, −8.051352337308495880478764667111, −7.35767416332621097285914793198, −5.98413065916960487951751915735, −5.19667268136125807114507816372, −3.75056949088258166882410190362, −2.72800026016992914882879677292, −1.66644659770392612638218462013,
1.66644659770392612638218462013, 2.72800026016992914882879677292, 3.75056949088258166882410190362, 5.19667268136125807114507816372, 5.98413065916960487951751915735, 7.35767416332621097285914793198, 8.051352337308495880478764667111, 8.900267729944967573194117141070, 9.721026942562876571917863787048, 10.28981135359909303761462495429
