L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s − 6·11-s − 4·13-s + 15-s + 2·17-s − 4·21-s + 23-s + 25-s − 27-s + 8·29-s + 8·31-s + 6·33-s − 4·35-s − 6·37-s + 4·39-s − 2·41-s + 6·43-s − 45-s − 12·47-s + 9·49-s − 2·51-s + 6·53-s + 6·55-s + 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.80·11-s − 1.10·13-s + 0.258·15-s + 0.485·17-s − 0.872·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.48·29-s + 1.43·31-s + 1.04·33-s − 0.676·35-s − 0.986·37-s + 0.640·39-s − 0.312·41-s + 0.914·43-s − 0.149·45-s − 1.75·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.809·55-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.297604899\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.297604899\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.452005795585591745855999831677, −8.042202953575078857225323508871, −7.47253487068690142520296301326, −6.61907519838549274842507322530, −5.34550243495374374598344940243, −5.02976444362114028256929415202, −4.42208819757340321897827140992, −2.99589463054103154686084114906, −2.10083273181205158420873033013, −0.72215829063851450553014284644,
0.72215829063851450553014284644, 2.10083273181205158420873033013, 2.99589463054103154686084114906, 4.42208819757340321897827140992, 5.02976444362114028256929415202, 5.34550243495374374598344940243, 6.61907519838549274842507322530, 7.47253487068690142520296301326, 8.042202953575078857225323508871, 8.452005795585591745855999831677