| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s + 9-s + 6·11-s − 12-s + 4·13-s + 4·14-s + 16-s − 6·17-s + 18-s − 4·21-s + 6·22-s − 6·23-s − 24-s + 4·26-s − 27-s + 4·28-s + 2·29-s + 32-s − 6·33-s − 6·34-s + 36-s + 8·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.288·12-s + 1.10·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.872·21-s + 1.27·22-s − 1.25·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.755·28-s + 0.371·29-s + 0.176·32-s − 1.04·33-s − 1.02·34-s + 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.346633897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.346633897\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.28749515840339, −13.99405438385538, −13.61640852255854, −12.76648968947045, −12.40280117772457, −11.76204056317030, −11.33520934649484, −11.07760368380487, −10.74529732509986, −9.788421651104785, −9.185554794873149, −8.746037168726061, −7.969441126526230, −7.704955736592940, −6.674993065753457, −6.455556042548972, −5.940997812253206, −5.324711043811508, −4.463394172907707, −4.224351065712164, −3.913435295491952, −2.793755260338931, −1.928644842088125, −1.486411764746331, −0.7864827926899421,
0.7864827926899421, 1.486411764746331, 1.928644842088125, 2.793755260338931, 3.913435295491952, 4.224351065712164, 4.463394172907707, 5.324711043811508, 5.940997812253206, 6.455556042548972, 6.674993065753457, 7.704955736592940, 7.969441126526230, 8.746037168726061, 9.185554794873149, 9.788421651104785, 10.74529732509986, 11.07760368380487, 11.33520934649484, 11.76204056317030, 12.40280117772457, 12.76648968947045, 13.61640852255854, 13.99405438385538, 14.28749515840339
