| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 4·7-s + 8-s + 9-s + 2·10-s + 4·11-s − 12-s + 13-s + 4·14-s − 2·15-s + 16-s + 4·17-s + 18-s − 19-s + 2·20-s − 4·21-s + 4·22-s − 24-s − 25-s + 26-s − 27-s + 4·28-s + 4·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.288·12-s + 0.277·13-s + 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.229·19-s + 0.447·20-s − 0.872·21-s + 0.852·22-s − 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.755·28-s + 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.523010958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.523010958\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.83132629114529, −13.30481759745172, −12.64142899137428, −12.22305708882749, −11.70372361376645, −11.45872562562168, −10.80538699938785, −10.48541528769051, −9.871147877212258, −9.270905303181111, −8.819469297583238, −8.132212035483231, −7.546216515698813, −7.198092381548327, −6.297057138786337, −6.059391049533104, −5.612648469463566, −4.993898333572448, −4.511563592598228, −4.047210245439276, −3.375831369471049, −2.506687963776939, −1.857213517276522, −1.374775404062519, −0.8505049648336486,
0.8505049648336486, 1.374775404062519, 1.857213517276522, 2.506687963776939, 3.375831369471049, 4.047210245439276, 4.511563592598228, 4.993898333572448, 5.612648469463566, 6.059391049533104, 6.297057138786337, 7.198092381548327, 7.546216515698813, 8.132212035483231, 8.819469297583238, 9.270905303181111, 9.871147877212258, 10.48541528769051, 10.80538699938785, 11.45872562562168, 11.70372361376645, 12.22305708882749, 12.64142899137428, 13.30481759745172, 13.83132629114529
