| L(s) = 1 | − 3-s − 5-s + 9-s − 6·13-s + 15-s + 2·17-s + 6·19-s − 23-s + 25-s − 27-s + 6·29-s − 4·31-s + 4·37-s + 6·39-s + 6·41-s + 4·43-s − 45-s − 4·47-s − 7·49-s − 2·51-s + 6·53-s − 6·57-s + 14·59-s + 6·61-s + 6·65-s − 4·67-s + 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.66·13-s + 0.258·15-s + 0.485·17-s + 1.37·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.657·37-s + 0.960·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 0.583·47-s − 49-s − 0.280·51-s + 0.824·53-s − 0.794·57-s + 1.82·59-s + 0.768·61-s + 0.744·65-s − 0.488·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.112463706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.112463706\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−9.830349159904273807842966310427, −8.825390991248242594074315530592, −7.62392559964906712855235250638, −7.37360111084066392691752490801, −6.27622844019593209092995838934, −5.27574267122710099342800580969, −4.68711930014682965123328517967, −3.53286821090762862082127588398, −2.40753305214926979883974253821, −0.78206889330272381020238582642,
0.78206889330272381020238582642, 2.40753305214926979883974253821, 3.53286821090762862082127588398, 4.68711930014682965123328517967, 5.27574267122710099342800580969, 6.27622844019593209092995838934, 7.37360111084066392691752490801, 7.62392559964906712855235250638, 8.825390991248242594074315530592, 9.830349159904273807842966310427
