L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 4·11-s + 12-s + 6·13-s + 15-s + 16-s + 2·17-s + 18-s + 4·19-s + 20-s − 4·22-s − 8·23-s + 24-s + 25-s + 6·26-s + 27-s + 6·29-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 + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 1.66·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.852·22-s − 1.66·23-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 − 0.179·31-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
\(L(1)\) |
\(\approx\) |
\(3.243778481\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.243778481\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−10.18699047326275641158873218687, −9.276619296306585781887542072454, −8.161080840551407359963888151910, −7.71700759612191021523492330172, −6.39041564261031743707990223277, −5.76525828987127075697992839640, −4.74518623784674296656955040749, −3.61176819345852177707332765191, −2.79033182625678861014177806881, −1.52532063559992491938859412184,
1.52532063559992491938859412184, 2.79033182625678861014177806881, 3.61176819345852177707332765191, 4.74518623784674296656955040749, 5.76525828987127075697992839640, 6.39041564261031743707990223277, 7.71700759612191021523492330172, 8.161080840551407359963888151910, 9.276619296306585781887542072454, 10.18699047326275641158873218687