| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 11-s + 12-s − 6·13-s − 15-s + 16-s − 17-s + 18-s − 4·19-s − 20-s − 22-s + 24-s + 25-s − 6·26-s + 27-s + 6·29-s − 30-s + 32-s − 33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 1.66·13-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.213·22-s + 0.204·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 1.11·29-s − 0.182·30-s + 0.176·32-s − 0.174·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 274890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 274890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.122417107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.122417107\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.66022808166631, −12.46119527303550, −11.95700914143604, −11.57699669411029, −10.91489919768669, −10.41242985019059, −10.17665104726779, −9.543943377269040, −9.025197152617627, −8.489679477840947, −8.105616942147128, −7.436060141226290, −7.240156656531577, −6.718829148321731, −6.129585315663758, −5.559325038612176, −4.902577923605721, −4.601830641959895, −4.131848278393630, −3.557113523910401, −2.925437510021112, −2.359619012300430, −2.215554047150286, −1.213446595951863, −0.3983039846025146,
0.3983039846025146, 1.213446595951863, 2.215554047150286, 2.359619012300430, 2.925437510021112, 3.557113523910401, 4.131848278393630, 4.601830641959895, 4.902577923605721, 5.559325038612176, 6.129585315663758, 6.718829148321731, 7.240156656531577, 7.436060141226290, 8.105616942147128, 8.489679477840947, 9.025197152617627, 9.543943377269040, 10.17665104726779, 10.41242985019059, 10.91489919768669, 11.57699669411029, 11.95700914143604, 12.46119527303550, 12.66022808166631
