L(s) = 1 | + 5-s − 2·7-s − 2·11-s + 2·13-s + 4·17-s + 6·19-s − 6·23-s + 25-s − 10·29-s − 8·31-s − 2·35-s − 2·37-s − 2·41-s + 43-s + 2·47-s − 3·49-s + 10·53-s − 2·55-s + 2·59-s + 12·61-s + 2·65-s + 12·67-s − 16·71-s + 16·73-s + 4·77-s − 8·79-s − 12·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.603·11-s + 0.554·13-s + 0.970·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s − 1.85·29-s − 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.312·41-s + 0.152·43-s + 0.291·47-s − 3/7·49-s + 1.37·53-s − 0.269·55-s + 0.260·59-s + 1.53·61-s + 0.248·65-s + 1.46·67-s − 1.89·71-s + 1.87·73-s + 0.455·77-s − 0.900·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.614548258\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.614548258\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.40630706214678, −13.11365733535403, −12.66068141406040, −12.17972108462965, −11.46984204915358, −11.28639056428319, −10.44022802597139, −10.12058542704739, −9.674232487598931, −9.252245082182790, −8.739193154080497, −7.950445317614411, −7.715121365856653, −7.006491946261169, −6.673940692210653, −5.694895845496284, −5.562638464399855, −5.324033365595927, −4.162819408167646, −3.696088211172650, −3.298564117233131, −2.578681444478959, −1.911533022433586, −1.285072102131984, −0.3886756584397880,
0.3886756584397880, 1.285072102131984, 1.911533022433586, 2.578681444478959, 3.298564117233131, 3.696088211172650, 4.162819408167646, 5.324033365595927, 5.562638464399855, 5.694895845496284, 6.673940692210653, 7.006491946261169, 7.715121365856653, 7.950445317614411, 8.739193154080497, 9.252245082182790, 9.674232487598931, 10.12058542704739, 10.44022802597139, 11.28639056428319, 11.46984204915358, 12.17972108462965, 12.66068141406040, 13.11365733535403, 13.40630706214678