L(s) = 1 | + 2.41·3-s − 0.585·5-s + 2.82·9-s − 11-s − 3.82·13-s − 1.41·15-s + 3.65·17-s + 0.585·19-s + 6.24·23-s − 4.65·25-s − 0.414·27-s + 2.65·29-s + 4·31-s − 2.41·33-s − 9.41·37-s − 9.24·39-s − 5.41·41-s + 5.65·43-s − 1.65·45-s + 10.4·47-s + 8.82·51-s + 7.89·53-s + 0.585·55-s + 1.41·57-s + 5.58·59-s + 11.8·61-s + 2.24·65-s + ⋯ |
L(s) = 1 | + 1.39·3-s − 0.261·5-s + 0.942·9-s − 0.301·11-s − 1.06·13-s − 0.365·15-s + 0.886·17-s + 0.134·19-s + 1.30·23-s − 0.931·25-s − 0.0797·27-s + 0.493·29-s + 0.718·31-s − 0.420·33-s − 1.54·37-s − 1.48·39-s − 0.845·41-s + 0.862·43-s − 0.246·45-s + 1.52·47-s + 1.23·51-s + 1.08·53-s + 0.0789·55-s + 0.187·57-s + 0.727·59-s + 1.51·61-s + 0.278·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.167987527\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.167987527\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 + 0.585T + 5T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 0.585T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 - 2.65T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 9.41T + 37T^{2} \) |
| 41 | \( 1 + 5.41T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 7.89T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 2.75T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 9.41T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 3.82T + 97T^{2} \) |
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
−7.74607036958580384323843468894, −7.35909382414697767814775652248, −6.67644969272791938895610773565, −5.50903971801855883138143774186, −4.99858711854964416335235212352, −3.99661333331763794510126724146, −3.40177385325507443005749039330, −2.65649552518627498150037535953, −2.06826634758642928235302460192, −0.799094839644138814639646614783,
0.799094839644138814639646614783, 2.06826634758642928235302460192, 2.65649552518627498150037535953, 3.40177385325507443005749039330, 3.99661333331763794510126724146, 4.99858711854964416335235212352, 5.50903971801855883138143774186, 6.67644969272791938895610773565, 7.35909382414697767814775652248, 7.74607036958580384323843468894