| L(s) = 1 | − 2-s − 3-s + 4-s − 4.16·5-s + 6-s − 8-s + 9-s + 4.16·10-s + 3.72·11-s − 12-s + 13-s + 4.16·15-s + 16-s − 2.59·17-s − 18-s − 2.37·19-s − 4.16·20-s − 3.72·22-s − 3.95·23-s + 24-s + 12.3·25-s − 26-s − 27-s − 4.84·29-s − 4.16·30-s − 0.227·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.86·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.31·10-s + 1.12·11-s − 0.288·12-s + 0.277·13-s + 1.07·15-s + 0.250·16-s − 0.630·17-s − 0.235·18-s − 0.543·19-s − 0.930·20-s − 0.793·22-s − 0.823·23-s + 0.204·24-s + 2.46·25-s − 0.196·26-s − 0.192·27-s − 0.898·29-s − 0.759·30-s − 0.0407·31-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
| \(L(1)\) |
\(\approx\) |
\(0.4639743221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4639743221\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4.16T + 5T^{2} \) |
| 11 | \( 1 - 3.72T + 11T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 23 | \( 1 + 3.95T + 23T^{2} \) |
| 29 | \( 1 + 4.84T + 29T^{2} \) |
| 31 | \( 1 + 0.227T + 31T^{2} \) |
| 37 | \( 1 - 1.04T + 37T^{2} \) |
| 41 | \( 1 + 2.35T + 41T^{2} \) |
| 43 | \( 1 - 7.95T + 43T^{2} \) |
| 47 | \( 1 - 2.17T + 47T^{2} \) |
| 53 | \( 1 + 4.17T + 53T^{2} \) |
| 59 | \( 1 - 7.54T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 6.56T + 67T^{2} \) |
| 71 | \( 1 + 7.88T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 17.9T + 83T^{2} \) |
| 89 | \( 1 + 4.79T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.506001781907634176670790857272, −7.74409986196196459663716599430, −7.14035601213164972318528616568, −6.52351083433112770800464114914, −5.68234362975248055282750267898, −4.28836009770559678441663874007, −4.13640727299736005861793620770, −3.07098457196835332911415531563, −1.64086507777731115750374196814, −0.45923803618539133672230390446,
0.45923803618539133672230390446, 1.64086507777731115750374196814, 3.07098457196835332911415531563, 4.13640727299736005861793620770, 4.28836009770559678441663874007, 5.68234362975248055282750267898, 6.52351083433112770800464114914, 7.14035601213164972318528616568, 7.74409986196196459663716599430, 8.506001781907634176670790857272
