| L(s) = 1 | + 2·3-s + 7-s + 9-s − 11-s − 6·17-s − 6·19-s + 2·21-s − 5·25-s − 4·27-s + 6·29-s − 2·33-s − 2·37-s + 2·41-s − 4·43-s + 49-s − 12·51-s − 6·53-s − 12·57-s − 2·59-s − 4·61-s + 63-s + 8·67-s − 16·71-s − 2·73-s − 10·75-s − 77-s − 8·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.45·17-s − 1.37·19-s + 0.436·21-s − 25-s − 0.769·27-s + 1.11·29-s − 0.348·33-s − 0.328·37-s + 0.312·41-s − 0.609·43-s + 1/7·49-s − 1.68·51-s − 0.824·53-s − 1.58·57-s − 0.260·59-s − 0.512·61-s + 0.125·63-s + 0.977·67-s − 1.89·71-s − 0.234·73-s − 1.15·75-s − 0.113·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−8.134764989537465931882037390317, −7.32842228704620224378494549501, −6.53165599388279635027968569329, −5.80096846617699863192462408090, −4.65706410759541849014615962900, −4.18814810955970701791190502224, −3.18225052077810574320306716298, −2.37976910318550006573655545086, −1.76009809097630322926645808384, 0,
1.76009809097630322926645808384, 2.37976910318550006573655545086, 3.18225052077810574320306716298, 4.18814810955970701791190502224, 4.65706410759541849014615962900, 5.80096846617699863192462408090, 6.53165599388279635027968569329, 7.32842228704620224378494549501, 8.134764989537465931882037390317
