| L(s) = 1 | − 3·3-s + 4·4-s + 11·7-s + 9·9-s − 12·12-s − 13-s + 16·16-s − 37·19-s − 33·21-s − 27·27-s + 44·28-s − 13·31-s + 36·36-s + 26·37-s + 3·39-s − 61·43-s − 48·48-s + 72·49-s − 4·52-s + 111·57-s + 47·61-s + 99·63-s + 64·64-s − 109·67-s − 46·73-s − 148·76-s − 142·79-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 11/7·7-s + 9-s − 12-s − 0.0769·13-s + 16-s − 1.94·19-s − 1.57·21-s − 27-s + 11/7·28-s − 0.419·31-s + 36-s + 0.702·37-s + 1/13·39-s − 1.41·43-s − 48-s + 1.46·49-s − 0.0769·52-s + 1.94·57-s + 0.770·61-s + 11/7·63-s + 64-s − 1.62·67-s − 0.630·73-s − 1.94·76-s − 1.79·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.271262568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.271262568\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 - 11 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 37 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 13 T + p^{2} T^{2} \) |
| 37 | \( 1 - 26 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 61 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 47 T + p^{2} T^{2} \) |
| 67 | \( 1 + 109 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 46 T + p^{2} T^{2} \) |
| 79 | \( 1 + 142 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 169 T + p^{2} T^{2} \) |
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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
−14.63292097656979508047831279792, −12.91310724151924088338631062947, −11.80237699260200269641583971921, −11.11846397779581478875527727995, −10.33330968458386745875603314991, −8.307330513157954818285381758515, −7.10668404379006521303995251976, −5.88202526001415638211028642709, −4.52562468114965560094721609275, −1.80688935404024327494295492875,
1.80688935404024327494295492875, 4.52562468114965560094721609275, 5.88202526001415638211028642709, 7.10668404379006521303995251976, 8.307330513157954818285381758515, 10.33330968458386745875603314991, 11.11846397779581478875527727995, 11.80237699260200269641583971921, 12.91310724151924088338631062947, 14.63292097656979508047831279792
