| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 2·7-s + 8-s + 9-s − 10-s + 12-s − 4·13-s + 2·14-s − 15-s + 16-s + 6·17-s + 18-s + 8·19-s − 20-s + 2·21-s + 24-s + 25-s − 4·26-s + 27-s + 2·28-s − 30-s + 31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 1.10·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 1.83·19-s − 0.223·20-s + 0.436·21-s + 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.377·28-s − 0.182·30-s + 0.179·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.043673922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.043673922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.01020655562293180863586044040, −9.356128742553520728301027733944, −7.930042068125277358317029520896, −7.76755724744181257433844878397, −6.77396254733465746129875816009, −5.35387075264389237189202605350, −4.86918753046544448344057980355, −3.63102841342813249977485855028, −2.85949719299743795484691919929, −1.44366684921523867007646091054,
1.44366684921523867007646091054, 2.85949719299743795484691919929, 3.63102841342813249977485855028, 4.86918753046544448344057980355, 5.35387075264389237189202605350, 6.77396254733465746129875816009, 7.76755724744181257433844878397, 7.930042068125277358317029520896, 9.356128742553520728301027733944, 10.01020655562293180863586044040
