| L(s) = 1 | − 3-s + 2·5-s + 4·7-s + 9-s + 13-s − 2·15-s − 17-s − 2·19-s − 4·21-s − 8·23-s − 25-s − 27-s − 2·29-s + 4·31-s + 8·35-s + 8·37-s − 39-s + 10·41-s + 2·45-s − 8·47-s + 9·49-s + 51-s − 2·53-s + 2·57-s − 6·61-s + 4·63-s + 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.516·15-s − 0.242·17-s − 0.458·19-s − 0.872·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 1.35·35-s + 1.31·37-s − 0.160·39-s + 1.56·41-s + 0.298·45-s − 1.16·47-s + 9/7·49-s + 0.140·51-s − 0.274·53-s + 0.264·57-s − 0.768·61-s + 0.503·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.272869106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.272869106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.42567500405820, −12.20916278058390, −11.66636121973856, −11.16464377841798, −10.89112882809236, −10.49111553334423, −9.874151839579496, −9.445418602038917, −9.146500656091921, −8.257852618813157, −7.933647444872601, −7.834734486897635, −6.930549367335189, −6.394213047228979, −5.990681962399778, −5.688618404479301, −5.040368685538649, −4.599704487899022, −4.196761749936330, −3.647713759106965, −2.704088108385258, −2.076814269665488, −1.861657159233084, −1.172973284902923, −0.5069787985010591,
0.5069787985010591, 1.172973284902923, 1.861657159233084, 2.076814269665488, 2.704088108385258, 3.647713759106965, 4.196761749936330, 4.599704487899022, 5.040368685538649, 5.688618404479301, 5.990681962399778, 6.394213047228979, 6.930549367335189, 7.834734486897635, 7.933647444872601, 8.257852618813157, 9.146500656091921, 9.445418602038917, 9.874151839579496, 10.49111553334423, 10.89112882809236, 11.16464377841798, 11.66636121973856, 12.20916278058390, 12.42567500405820
