| L(s) = 1 | + 2·2-s + 4·4-s + 8·8-s − 14·11-s + 16·16-s − 2·17-s − 34·19-s − 28·22-s + 25·25-s + 32·32-s − 4·34-s − 68·38-s + 46·41-s + 14·43-s − 56·44-s + 49·49-s + 50·50-s + 82·59-s + 64·64-s + 62·67-s − 8·68-s − 142·73-s − 136·76-s + 92·82-s − 158·83-s + 28·86-s − 112·88-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 8-s − 1.27·11-s + 16-s − 0.117·17-s − 1.78·19-s − 1.27·22-s + 25-s + 32-s − 0.117·34-s − 1.78·38-s + 1.12·41-s + 0.325·43-s − 1.27·44-s + 49-s + 50-s + 1.38·59-s + 64-s + 0.925·67-s − 0.117·68-s − 1.94·73-s − 1.78·76-s + 1.12·82-s − 1.90·83-s + 0.325·86-s − 1.27·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.055769513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.055769513\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 + 14 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 + 2 T + p^{2} T^{2} \) |
| 19 | \( 1 + 34 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( 1 - 46 T + p^{2} T^{2} \) |
| 43 | \( 1 - 14 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( 1 - 82 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 - 62 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 142 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( 1 + 158 T + p^{2} T^{2} \) |
| 89 | \( 1 + 146 T + p^{2} T^{2} \) |
| 97 | \( 1 + 94 T + p^{2} T^{2} \) |
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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
−14.39569596648933960311916381740, −13.14693378937286856306913915897, −12.57046798682164528150473448557, −11.12222283902763454964896200586, −10.31668919882836182511271258838, −8.406498133634436240111534652661, −7.06031335001717125662337247420, −5.70544546324882239535508589477, −4.36754048222466593649841531068, −2.55815955809632910426033901254,
2.55815955809632910426033901254, 4.36754048222466593649841531068, 5.70544546324882239535508589477, 7.06031335001717125662337247420, 8.406498133634436240111534652661, 10.31668919882836182511271258838, 11.12222283902763454964896200586, 12.57046798682164528150473448557, 13.14693378937286856306913915897, 14.39569596648933960311916381740
