| L(s) = 1 | + 3-s − 2·5-s − 2·9-s − 11-s − 3·13-s − 2·15-s + 2·17-s − 4·19-s − 4·23-s − 25-s − 5·27-s − 7·29-s + 8·31-s − 33-s + 12·37-s − 3·39-s + 8·41-s + 8·43-s + 4·45-s + 10·47-s + 2·51-s + 14·53-s + 2·55-s − 4·57-s + 9·59-s + 5·61-s + 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 2/3·9-s − 0.301·11-s − 0.832·13-s − 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 0.962·27-s − 1.29·29-s + 1.43·31-s − 0.174·33-s + 1.97·37-s − 0.480·39-s + 1.24·41-s + 1.21·43-s + 0.596·45-s + 1.45·47-s + 0.280·51-s + 1.92·53-s + 0.269·55-s − 0.529·57-s + 1.17·59-s + 0.640·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.313050527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.313050527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.274784960264415587066415518968, −7.69619783897789859252090965617, −7.28472223386266498109554938445, −6.03641681048224256732962309097, −5.56370101142428966790643480266, −4.24517946032826634221867027709, −4.01178895413176674727905474738, −2.75211145609864687659993898118, −2.31413349926768417841529967001, −0.59640210093871342472667399884,
0.59640210093871342472667399884, 2.31413349926768417841529967001, 2.75211145609864687659993898118, 4.01178895413176674727905474738, 4.24517946032826634221867027709, 5.56370101142428966790643480266, 6.03641681048224256732962309097, 7.28472223386266498109554938445, 7.69619783897789859252090965617, 8.274784960264415587066415518968
