L(s) = 1 | + 5-s − 5·11-s + 2·13-s − 6·17-s + 2·19-s + 6·23-s − 4·25-s + 3·29-s + 5·31-s − 2·37-s − 8·41-s − 4·43-s − 4·47-s − 9·53-s − 5·55-s + 3·59-s − 12·61-s + 2·65-s + 2·67-s + 8·71-s − 14·73-s + 79-s − 17·83-s − 6·85-s − 18·89-s + 2·95-s + 3·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.50·11-s + 0.554·13-s − 1.45·17-s + 0.458·19-s + 1.25·23-s − 4/5·25-s + 0.557·29-s + 0.898·31-s − 0.328·37-s − 1.24·41-s − 0.609·43-s − 0.583·47-s − 1.23·53-s − 0.674·55-s + 0.390·59-s − 1.53·61-s + 0.248·65-s + 0.244·67-s + 0.949·71-s − 1.63·73-s + 0.112·79-s − 1.86·83-s − 0.650·85-s − 1.90·89-s + 0.205·95-s + 0.304·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−8.337516670272648119605602855957, −7.45043598334643694649710246650, −6.68591860253959306152205905130, −5.96931823241425137796119471289, −5.08880105209591815606035711471, −4.56411109885534067610717464934, −3.28450940381689600827697026965, −2.57473908479177302998102945671, −1.53616741177806075948287103933, 0,
1.53616741177806075948287103933, 2.57473908479177302998102945671, 3.28450940381689600827697026965, 4.56411109885534067610717464934, 5.08880105209591815606035711471, 5.96931823241425137796119471289, 6.68591860253959306152205905130, 7.45043598334643694649710246650, 8.337516670272648119605602855957