| L(s) = 1 | + 3-s + 4·4-s − 5·5-s + 7·7-s − 8·9-s − 13·11-s + 4·12-s − 19·13-s − 5·15-s + 16·16-s + 29·17-s − 20·20-s + 7·21-s + 25·25-s − 17·27-s + 28·28-s + 23·29-s − 13·33-s − 35·35-s − 32·36-s − 19·39-s − 52·44-s + 40·45-s − 31·47-s + 16·48-s + 49·49-s + 29·51-s + ⋯ |
| L(s) = 1 | + 1/3·3-s + 4-s − 5-s + 7-s − 8/9·9-s − 1.18·11-s + 1/3·12-s − 1.46·13-s − 1/3·15-s + 16-s + 1.70·17-s − 20-s + 1/3·21-s + 25-s − 0.629·27-s + 28-s + 0.793·29-s − 0.393·33-s − 35-s − 8/9·36-s − 0.487·39-s − 1.18·44-s + 8/9·45-s − 0.659·47-s + 1/3·48-s + 49-s + 0.568·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.156189900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.156189900\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 3 | \( 1 - T + p^{2} T^{2} \) |
| 11 | \( 1 + 13 T + p^{2} T^{2} \) |
| 13 | \( 1 + 19 T + p^{2} T^{2} \) |
| 17 | \( 1 - 29 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 - 23 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 + 31 T + p^{2} T^{2} \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( 1 - 2 T + p^{2} T^{2} \) |
| 73 | \( 1 + 34 T + p^{2} T^{2} \) |
| 79 | \( 1 + 157 T + p^{2} T^{2} \) |
| 83 | \( 1 - 86 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 149 T + p^{2} T^{2} \) |
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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
−16.23116852344103051500633320946, −15.02607007700195946026435981541, −14.41633962392969628700169423378, −12.32310638626126060564211726011, −11.54972439439371729211233968125, −10.33763642089133302017548153955, −8.122254405645620978061782440328, −7.50825470244150866666977988097, −5.24975339700447806875919731332, −2.84555816738779478639418436323,
2.84555816738779478639418436323, 5.24975339700447806875919731332, 7.50825470244150866666977988097, 8.122254405645620978061782440328, 10.33763642089133302017548153955, 11.54972439439371729211233968125, 12.32310638626126060564211726011, 14.41633962392969628700169423378, 15.02607007700195946026435981541, 16.23116852344103051500633320946
