| L(s) = 1 | + 3·3-s + 16·7-s + 9·9-s − 28·11-s + 26·13-s + 62·17-s − 68·19-s + 48·21-s + 208·23-s + 27·27-s − 58·29-s + 160·31-s − 84·33-s − 270·37-s + 78·39-s + 282·41-s − 76·43-s + 280·47-s − 87·49-s + 186·51-s + 210·53-s − 204·57-s + 196·59-s + 742·61-s + 144·63-s − 836·67-s + 624·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.863·7-s + 1/3·9-s − 0.767·11-s + 0.554·13-s + 0.884·17-s − 0.821·19-s + 0.498·21-s + 1.88·23-s + 0.192·27-s − 0.371·29-s + 0.926·31-s − 0.443·33-s − 1.19·37-s + 0.320·39-s + 1.07·41-s − 0.269·43-s + 0.868·47-s − 0.253·49-s + 0.510·51-s + 0.544·53-s − 0.474·57-s + 0.432·59-s + 1.55·61-s + 0.287·63-s − 1.52·67-s + 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.843549877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.843549877\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 62 T + p^{3} T^{2} \) |
| 19 | \( 1 + 68 T + p^{3} T^{2} \) |
| 23 | \( 1 - 208 T + p^{3} T^{2} \) |
| 29 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 160 T + p^{3} T^{2} \) |
| 37 | \( 1 + 270 T + p^{3} T^{2} \) |
| 41 | \( 1 - 282 T + p^{3} T^{2} \) |
| 43 | \( 1 + 76 T + p^{3} T^{2} \) |
| 47 | \( 1 - 280 T + p^{3} T^{2} \) |
| 53 | \( 1 - 210 T + p^{3} T^{2} \) |
| 59 | \( 1 - 196 T + p^{3} T^{2} \) |
| 61 | \( 1 - 742 T + p^{3} T^{2} \) |
| 67 | \( 1 + 836 T + p^{3} T^{2} \) |
| 71 | \( 1 + 504 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1062 T + p^{3} T^{2} \) |
| 79 | \( 1 - 768 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1052 T + p^{3} T^{2} \) |
| 89 | \( 1 + 726 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1406 T + p^{3} T^{2} \) |
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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
−10.40315258013460502497575714427, −9.231469528224753965378959459148, −8.449190075546532196723925377173, −7.77635714893283745088183976587, −6.83701489903676173234531684575, −5.53609398247293713209504942330, −4.67871952250030086479151753184, −3.47048686458413917204240061837, −2.34248148855115171478782764637, −1.04256043740078889423438437899,
1.04256043740078889423438437899, 2.34248148855115171478782764637, 3.47048686458413917204240061837, 4.67871952250030086479151753184, 5.53609398247293713209504942330, 6.83701489903676173234531684575, 7.77635714893283745088183976587, 8.449190075546532196723925377173, 9.231469528224753965378959459148, 10.40315258013460502497575714427
