L(s) = 1 | − 2-s + 4-s − 3·7-s − 8-s − 3·9-s − 13-s + 3·14-s + 16-s − 5·17-s + 3·18-s − 19-s + 5·23-s − 5·25-s + 26-s − 3·28-s − 6·29-s + 2·31-s − 32-s + 5·34-s − 3·36-s − 3·37-s + 38-s + 2·41-s + 4·43-s − 5·46-s − 3·47-s + 2·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.13·7-s − 0.353·8-s − 9-s − 0.277·13-s + 0.801·14-s + 1/4·16-s − 1.21·17-s + 0.707·18-s − 0.229·19-s + 1.04·23-s − 25-s + 0.196·26-s − 0.566·28-s − 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.857·34-s − 1/2·36-s − 0.493·37-s + 0.162·38-s + 0.312·41-s + 0.609·43-s − 0.737·46-s − 0.437·47-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4907690306\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4907690306\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.581667113096120600802336877701, −7.54918850751004911454935889811, −6.97155050259421063428873086985, −6.18461518574718440667927582930, −5.68875401236409707623437779186, −4.58579598877322811410162405584, −3.52550885482380618682257711478, −2.80583407766484257278858832428, −1.97296683610578027538137202597, −0.40520920389872484228368834466,
0.40520920389872484228368834466, 1.97296683610578027538137202597, 2.80583407766484257278858832428, 3.52550885482380618682257711478, 4.58579598877322811410162405584, 5.68875401236409707623437779186, 6.18461518574718440667927582930, 6.97155050259421063428873086985, 7.54918850751004911454935889811, 8.581667113096120600802336877701