| L(s) = 1 | + 3-s + 9-s − 2·11-s + 2·13-s − 4·17-s + 4·19-s − 6·23-s − 5·25-s + 27-s − 2·29-s − 2·33-s − 6·37-s + 2·39-s − 8·41-s − 8·43-s + 4·47-s − 4·51-s − 6·53-s + 4·57-s + 14·61-s + 4·67-s − 6·69-s − 2·71-s + 2·73-s − 5·75-s + 4·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.970·17-s + 0.917·19-s − 1.25·23-s − 25-s + 0.192·27-s − 0.371·29-s − 0.348·33-s − 0.986·37-s + 0.320·39-s − 1.24·41-s − 1.21·43-s + 0.583·47-s − 0.560·51-s − 0.824·53-s + 0.529·57-s + 1.79·61-s + 0.488·67-s − 0.722·69-s − 0.237·71-s + 0.234·73-s − 0.577·75-s + 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.129741957826259213723723250066, −7.26620416214267805722774714674, −6.59717357681308295714693544426, −5.70258321188560248598095913524, −4.99890635713642333126858923816, −3.99308774084127825312642725754, −3.41118252241251894008241546348, −2.36786946563000943133137823692, −1.59532486403730750473684940470, 0,
1.59532486403730750473684940470, 2.36786946563000943133137823692, 3.41118252241251894008241546348, 3.99308774084127825312642725754, 4.99890635713642333126858923816, 5.70258321188560248598095913524, 6.59717357681308295714693544426, 7.26620416214267805722774714674, 8.129741957826259213723723250066
