L(s) = 1 | − 5-s + 7-s + 5·11-s − 5·13-s + 7·17-s − 2·19-s + 2·23-s + 25-s − 7·29-s + 4·31-s − 35-s − 6·37-s + 12·41-s − 2·43-s − 47-s + 49-s − 5·55-s + 4·59-s + 4·61-s + 5·65-s + 8·67-s + 6·73-s + 5·77-s − 3·79-s + 4·83-s − 7·85-s − 5·91-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.50·11-s − 1.38·13-s + 1.69·17-s − 0.458·19-s + 0.417·23-s + 1/5·25-s − 1.29·29-s + 0.718·31-s − 0.169·35-s − 0.986·37-s + 1.87·41-s − 0.304·43-s − 0.145·47-s + 1/7·49-s − 0.674·55-s + 0.520·59-s + 0.512·61-s + 0.620·65-s + 0.977·67-s + 0.702·73-s + 0.569·77-s − 0.337·79-s + 0.439·83-s − 0.759·85-s − 0.524·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.831603530\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.831603530\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−8.986591679604333601542857191321, −8.033082332859894178672793368914, −7.42695240476731979776224139494, −6.74907559299019305057473208578, −5.74978286836860489420343080410, −4.93936350214085093868248089925, −4.06954744861732417689986542370, −3.30527145306019394907962235588, −2.07628009905382843609630348286, −0.887538960576224120433585893779,
0.887538960576224120433585893779, 2.07628009905382843609630348286, 3.30527145306019394907962235588, 4.06954744861732417689986542370, 4.93936350214085093868248089925, 5.74978286836860489420343080410, 6.74907559299019305057473208578, 7.42695240476731979776224139494, 8.033082332859894178672793368914, 8.986591679604333601542857191321