L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 5·11-s + 12-s + 13-s − 15-s + 16-s + 3·17-s − 18-s + 5·19-s − 20-s − 5·22-s + 9·23-s − 24-s − 4·25-s − 26-s + 27-s − 29-s + 30-s − 2·31-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 − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 1.14·19-s − 0.223·20-s − 1.06·22-s + 1.87·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s − 0.185·29-s + 0.182·30-s − 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.919684739\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.919684739\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.591723455201273460210612888316, −7.77500294269523198123673543627, −7.22373511360714970789561878448, −6.55658578888365028760636975760, −5.61904536148747937937631714882, −4.59327506435199313363552389012, −3.54060660250747025836426317323, −3.14030157951736944573725569572, −1.72493460420431645394751626661, −0.942361798317428019869143990447,
0.942361798317428019869143990447, 1.72493460420431645394751626661, 3.14030157951736944573725569572, 3.54060660250747025836426317323, 4.59327506435199313363552389012, 5.61904536148747937937631714882, 6.55658578888365028760636975760, 7.22373511360714970789561878448, 7.77500294269523198123673543627, 8.591723455201273460210612888316