| L(s) = 1 | − 3-s + 7-s − 2·9-s + 11-s − 17-s − 19-s − 21-s + 5·27-s − 29-s + 31-s − 33-s − 37-s + 6·43-s + 8·47-s − 6·49-s + 51-s − 9·53-s + 57-s − 4·59-s − 7·61-s − 2·63-s − 4·67-s − 5·71-s − 14·73-s + 77-s − 4·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.301·11-s − 0.242·17-s − 0.229·19-s − 0.218·21-s + 0.962·27-s − 0.185·29-s + 0.179·31-s − 0.174·33-s − 0.164·37-s + 0.914·43-s + 1.16·47-s − 6/7·49-s + 0.140·51-s − 1.23·53-s + 0.132·57-s − 0.520·59-s − 0.896·61-s − 0.251·63-s − 0.488·67-s − 0.593·71-s − 1.63·73-s + 0.113·77-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.951662380660049215969666649110, −7.29997142715825538555598519161, −6.31897820911954060047884905099, −5.92559031307278184228698268950, −5.00662765949998344422587714046, −4.39602918659409123463354737339, −3.35707500162431374995492464731, −2.42985027556468358842264902413, −1.29861350723040713076403758694, 0,
1.29861350723040713076403758694, 2.42985027556468358842264902413, 3.35707500162431374995492464731, 4.39602918659409123463354737339, 5.00662765949998344422587714046, 5.92559031307278184228698268950, 6.31897820911954060047884905099, 7.29997142715825538555598519161, 7.951662380660049215969666649110
