| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s − 4·7-s + 8-s + 9-s − 2·10-s + 12-s − 2·13-s − 4·14-s − 2·15-s + 16-s + 6·17-s + 18-s + 4·19-s − 2·20-s − 4·21-s − 23-s + 24-s − 25-s − 2·26-s + 27-s − 4·28-s + 29-s − 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s − 0.554·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.872·21-s − 0.208·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.755·28-s + 0.185·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.582431080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.582431080\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.117923352354560842576493810952, −7.72241330075962692237819063696, −7.01333323419057346375191818255, −6.24812917471354955721910234661, −5.46356916712438031826540105298, −4.50162287690062263138037116263, −3.55784116503777040358801542907, −3.29995376479477247004032977433, −2.38745920662164929006825210076, −0.792012704452018534737806150565,
0.792012704452018534737806150565, 2.38745920662164929006825210076, 3.29995376479477247004032977433, 3.55784116503777040358801542907, 4.50162287690062263138037116263, 5.46356916712438031826540105298, 6.24812917471354955721910234661, 7.01333323419057346375191818255, 7.72241330075962692237819063696, 8.117923352354560842576493810952
