L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s − 3·11-s − 12-s + 13-s − 2·14-s + 16-s − 3·17-s − 18-s − 5·19-s − 2·21-s + 3·22-s − 4·23-s + 24-s − 26-s − 27-s + 2·28-s + 31-s − 32-s + 3·33-s + 3·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 1.14·19-s − 0.436·21-s + 0.639·22-s − 0.834·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.377·28-s + 0.179·31-s − 0.176·32-s + 0.522·33-s + 0.514·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8733623881\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8733623881\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.134414030998571150140205387151, −7.85836139350364969661135442454, −6.85326032220411625879289729562, −6.23800101454284532986876834508, −5.46353472214034969877940876269, −4.66312714621253583918676828886, −3.89093857435695056714393357545, −2.52551460805205806062413382859, −1.85633600665032330625693437133, −0.58248240546343598230121795365,
0.58248240546343598230121795365, 1.85633600665032330625693437133, 2.52551460805205806062413382859, 3.89093857435695056714393357545, 4.66312714621253583918676828886, 5.46353472214034969877940876269, 6.23800101454284532986876834508, 6.85326032220411625879289729562, 7.85836139350364969661135442454, 8.134414030998571150140205387151