L(s) = 1 | − 1.01·3-s + 3.66·5-s − 1.84·7-s − 1.97·9-s − 2.84·11-s + 4.78·13-s − 3.70·15-s − 7.85·17-s − 1.81·19-s + 1.87·21-s + 5.67·23-s + 8.42·25-s + 5.03·27-s + 7.12·29-s + 2.60·31-s + 2.87·33-s − 6.77·35-s + 4.12·37-s − 4.84·39-s + 10.0·41-s − 4.94·43-s − 7.24·45-s − 10.9·47-s − 3.57·49-s + 7.94·51-s − 12.8·53-s − 10.4·55-s + ⋯ |
L(s) = 1 | − 0.584·3-s + 1.63·5-s − 0.699·7-s − 0.658·9-s − 0.857·11-s + 1.32·13-s − 0.957·15-s − 1.90·17-s − 0.415·19-s + 0.408·21-s + 1.18·23-s + 1.68·25-s + 0.969·27-s + 1.32·29-s + 0.467·31-s + 0.501·33-s − 1.14·35-s + 0.678·37-s − 0.775·39-s + 1.56·41-s − 0.754·43-s − 1.07·45-s − 1.59·47-s − 0.511·49-s + 1.11·51-s − 1.76·53-s − 1.40·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.696246760\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.696246760\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + 1.01T + 3T^{2} \) |
| 5 | \( 1 - 3.66T + 5T^{2} \) |
| 7 | \( 1 + 1.84T + 7T^{2} \) |
| 11 | \( 1 + 2.84T + 11T^{2} \) |
| 13 | \( 1 - 4.78T + 13T^{2} \) |
| 17 | \( 1 + 7.85T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 - 5.67T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 - 2.60T + 31T^{2} \) |
| 37 | \( 1 - 4.12T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 4.94T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 + 1.38T + 61T^{2} \) |
| 67 | \( 1 - 6.14T + 67T^{2} \) |
| 71 | \( 1 + 2.57T + 71T^{2} \) |
| 73 | \( 1 - 5.14T + 73T^{2} \) |
| 79 | \( 1 - 7.49T + 79T^{2} \) |
| 83 | \( 1 - 8.68T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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
−7.925741284385569857824348708033, −6.57639356826632588510512681546, −6.34118805916693114059092298357, −6.08902082816677695283327807146, −4.98811950794480011414109689267, −4.69881714031171568804487299687, −3.21784110048327744652560693102, −2.67758698182346674603039738496, −1.81454262812173934754714403699, −0.65016527894844670338850132450,
0.65016527894844670338850132450, 1.81454262812173934754714403699, 2.67758698182346674603039738496, 3.21784110048327744652560693102, 4.69881714031171568804487299687, 4.98811950794480011414109689267, 6.08902082816677695283327807146, 6.34118805916693114059092298357, 6.57639356826632588510512681546, 7.925741284385569857824348708033