L(s) = 1 | − 3-s + 3.76·5-s + 1.84·7-s + 9-s + 1.01·11-s + 4.04·13-s − 3.76·15-s − 1.59·17-s − 1.49·19-s − 1.84·21-s + 8.43·23-s + 9.19·25-s − 27-s + 0.429·29-s + 3.08·31-s − 1.01·33-s + 6.95·35-s + 4.40·37-s − 4.04·39-s − 5.20·41-s − 8.83·43-s + 3.76·45-s − 4.52·47-s − 3.59·49-s + 1.59·51-s + 11.5·53-s + 3.83·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.68·5-s + 0.697·7-s + 0.333·9-s + 0.306·11-s + 1.12·13-s − 0.972·15-s − 0.386·17-s − 0.343·19-s − 0.402·21-s + 1.75·23-s + 1.83·25-s − 0.192·27-s + 0.0797·29-s + 0.554·31-s − 0.177·33-s + 1.17·35-s + 0.723·37-s − 0.647·39-s − 0.813·41-s − 1.34·43-s + 0.561·45-s − 0.659·47-s − 0.513·49-s + 0.223·51-s + 1.58·53-s + 0.516·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.995136700\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.995136700\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
good | 5 | \( 1 - 3.76T + 5T^{2} \) |
| 7 | \( 1 - 1.84T + 7T^{2} \) |
| 11 | \( 1 - 1.01T + 11T^{2} \) |
| 13 | \( 1 - 4.04T + 13T^{2} \) |
| 17 | \( 1 + 1.59T + 17T^{2} \) |
| 19 | \( 1 + 1.49T + 19T^{2} \) |
| 23 | \( 1 - 8.43T + 23T^{2} \) |
| 29 | \( 1 - 0.429T + 29T^{2} \) |
| 31 | \( 1 - 3.08T + 31T^{2} \) |
| 37 | \( 1 - 4.40T + 37T^{2} \) |
| 41 | \( 1 + 5.20T + 41T^{2} \) |
| 43 | \( 1 + 8.83T + 43T^{2} \) |
| 47 | \( 1 + 4.52T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 5.83T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 0.0872T + 71T^{2} \) |
| 73 | \( 1 - 9.41T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 5.38T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−8.307431256460753518681917994535, −7.07094723779879153931465189165, −6.49264534609489081614517916689, −6.04004299903550089462023984043, −5.13825627924291544955799469057, −4.83215458504931405364228845544, −3.65815342966886659946228495268, −2.59028313394131813430817073987, −1.66589243005847510347192155768, −1.05002211511824257549310848198,
1.05002211511824257549310848198, 1.66589243005847510347192155768, 2.59028313394131813430817073987, 3.65815342966886659946228495268, 4.83215458504931405364228845544, 5.13825627924291544955799469057, 6.04004299903550089462023984043, 6.49264534609489081614517916689, 7.07094723779879153931465189165, 8.307431256460753518681917994535