| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s + 7-s + 8-s + 9-s − 2·10-s + 4·11-s + 12-s − 6·13-s + 14-s − 2·15-s + 16-s − 2·17-s + 18-s + 4·19-s − 2·20-s + 21-s + 4·22-s + 8·23-s + 24-s − 25-s − 6·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.218·21-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 + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70602 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70602 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.342189629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.342189629\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 41 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.20235755143753, −13.84228704847204, −13.02433626786762, −12.72466768141295, −12.01676749562516, −11.67771890476836, −11.50235911500948, −10.63046205953682, −10.16856698546718, −9.446011170326138, −9.056686771573416, −8.504220269194146, −7.786291140575981, −7.412376575656027, −6.881930891642296, −6.596218545841352, −5.543018318353712, −4.941863069366285, −4.653322810057570, −3.962560423705013, −3.391265514578200, −2.948012423940121, −2.145454178876078, −1.496647579968130, −0.5901047702660019,
0.5901047702660019, 1.496647579968130, 2.145454178876078, 2.948012423940121, 3.391265514578200, 3.962560423705013, 4.653322810057570, 4.941863069366285, 5.543018318353712, 6.596218545841352, 6.881930891642296, 7.412376575656027, 7.786291140575981, 8.504220269194146, 9.056686771573416, 9.446011170326138, 10.16856698546718, 10.63046205953682, 11.50235911500948, 11.67771890476836, 12.01676749562516, 12.72466768141295, 13.02433626786762, 13.84228704847204, 14.20235755143753
