| L(s) = 1 | + 2·5-s + 4·7-s + 2·11-s − 5·13-s − 4·17-s + 2·19-s + 23-s − 25-s + 7·29-s + 3·31-s + 8·35-s + 2·37-s + 9·41-s + 8·43-s + 9·47-s + 9·49-s − 2·53-s + 4·55-s − 2·61-s − 10·65-s − 14·67-s − 3·71-s − 3·73-s + 8·77-s + 6·79-s + 8·83-s − 8·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s + 0.603·11-s − 1.38·13-s − 0.970·17-s + 0.458·19-s + 0.208·23-s − 1/5·25-s + 1.29·29-s + 0.538·31-s + 1.35·35-s + 0.328·37-s + 1.40·41-s + 1.21·43-s + 1.31·47-s + 9/7·49-s − 0.274·53-s + 0.539·55-s − 0.256·61-s − 1.24·65-s − 1.71·67-s − 0.356·71-s − 0.351·73-s + 0.911·77-s + 0.675·79-s + 0.878·83-s − 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.717638386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.717638386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.762474026916601727659642248899, −7.78431087011820196153358198018, −7.28380268960996565959039707529, −6.30231354857361067552946600683, −5.58252703725131255817507716797, −4.68687938596758625929687535386, −4.32763445735159785512127137050, −2.70442367409739251308486042341, −2.08210841090047127789623388388, −1.04926008910287368677124435543,
1.04926008910287368677124435543, 2.08210841090047127789623388388, 2.70442367409739251308486042341, 4.32763445735159785512127137050, 4.68687938596758625929687535386, 5.58252703725131255817507716797, 6.30231354857361067552946600683, 7.28380268960996565959039707529, 7.78431087011820196153358198018, 8.762474026916601727659642248899
