L(s) = 1 | − 3-s − 5-s + 9-s + 15-s + 2·17-s + 23-s + 25-s − 27-s + 6·29-s − 8·31-s + 10·37-s − 2·43-s − 45-s + 6·47-s − 2·51-s + 14·53-s − 14·59-s − 6·61-s − 10·67-s − 69-s + 12·71-s − 14·73-s − 75-s + 14·79-s + 81-s + 12·83-s − 2·85-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.258·15-s + 0.485·17-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.304·43-s − 0.149·45-s + 0.875·47-s − 0.280·51-s + 1.92·53-s − 1.82·59-s − 0.768·61-s − 1.22·67-s − 0.120·69-s + 1.42·71-s − 1.63·73-s − 0.115·75-s + 1.57·79-s + 1/9·81-s + 1.31·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.870417441\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.870417441\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.54143008078493, −12.80398065006771, −12.41638963951000, −12.03925951229801, −11.56357327255287, −11.01448891718430, −10.66180784431233, −10.16262855568714, −9.623616693438618, −9.018005106530217, −8.678523007028050, −7.906707496569803, −7.491460014918686, −7.177682531718978, −6.353846510553764, −6.047128868898237, −5.475762122522045, −4.792545763934950, −4.480058790417704, −3.752141072832898, −3.264643023869544, −2.559138762211357, −1.856706480377237, −1.022468848125737, −0.5053665989192457,
0.5053665989192457, 1.022468848125737, 1.856706480377237, 2.559138762211357, 3.264643023869544, 3.752141072832898, 4.480058790417704, 4.792545763934950, 5.475762122522045, 6.047128868898237, 6.353846510553764, 7.177682531718978, 7.491460014918686, 7.906707496569803, 8.678523007028050, 9.018005106530217, 9.623616693438618, 10.16262855568714, 10.66180784431233, 11.01448891718430, 11.56357327255287, 12.03925951229801, 12.41638963951000, 12.80398065006771, 13.54143008078493