| L(s) = 1 | + 3-s + 0.585·5-s + 9-s − 0.828·11-s − 1.41·13-s + 0.585·15-s + 2.24·17-s + 6.82·19-s + 4.82·23-s − 4.65·25-s + 27-s + 8.48·29-s + 5.17·31-s − 0.828·33-s + 1.65·37-s − 1.41·39-s − 0.585·41-s − 8·43-s + 0.585·45-s + 6.82·47-s + 2.24·51-s − 13.3·53-s − 0.485·55-s + 6.82·57-s + 5.17·59-s + 13.8·61-s − 0.828·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.261·5-s + 0.333·9-s − 0.249·11-s − 0.392·13-s + 0.151·15-s + 0.543·17-s + 1.56·19-s + 1.00·23-s − 0.931·25-s + 0.192·27-s + 1.57·29-s + 0.928·31-s − 0.144·33-s + 0.272·37-s − 0.226·39-s − 0.0914·41-s − 1.21·43-s + 0.0873·45-s + 0.996·47-s + 0.314·51-s − 1.82·53-s − 0.0654·55-s + 0.904·57-s + 0.673·59-s + 1.77·61-s − 0.102·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.191811069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.191811069\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 5.17T + 31T^{2} \) |
| 37 | \( 1 - 1.65T + 37T^{2} \) |
| 41 | \( 1 + 0.585T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 6.82T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 0.828T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 7.75T + 97T^{2} \) |
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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
−9.861077245890059112473975054621, −8.988043505261851823510339518785, −8.101603910814072625241201013756, −7.43237885148402138107504187746, −6.53073269406908781822797002563, −5.43706643098702948040279093043, −4.65139888895648503476376301733, −3.37231425399933199119704063175, −2.60595467631557241019258042097, −1.18230171322547453703821044986,
1.18230171322547453703821044986, 2.60595467631557241019258042097, 3.37231425399933199119704063175, 4.65139888895648503476376301733, 5.43706643098702948040279093043, 6.53073269406908781822797002563, 7.43237885148402138107504187746, 8.101603910814072625241201013756, 8.988043505261851823510339518785, 9.861077245890059112473975054621
