L(s) = 1 | + 2-s − 0.445·3-s + 4-s − 0.445·6-s + 8-s − 0.801·9-s + 1.24·11-s − 0.445·12-s + 16-s + 1.24·17-s − 0.801·18-s − 1.80·19-s + 1.24·22-s − 0.445·24-s + 25-s + 0.801·27-s + 32-s − 0.554·33-s + 1.24·34-s − 0.801·36-s − 1.80·38-s − 1.80·41-s − 1.80·43-s + 1.24·44-s − 0.445·48-s + 49-s + 50-s + ⋯ |
L(s) = 1 | + 2-s − 0.445·3-s + 4-s − 0.445·6-s + 8-s − 0.801·9-s + 1.24·11-s − 0.445·12-s + 16-s + 1.24·17-s − 0.801·18-s − 1.80·19-s + 1.24·22-s − 0.445·24-s + 25-s + 0.801·27-s + 32-s − 0.554·33-s + 1.24·34-s − 0.801·36-s − 1.80·38-s − 1.80·41-s − 1.80·43-s + 1.24·44-s − 0.445·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.787745126\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.787745126\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 0.445T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.24T + T^{2} \) |
| 17 | \( 1 - 1.24T + T^{2} \) |
| 19 | \( 1 + 1.80T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.80T + T^{2} \) |
| 43 | \( 1 + 1.80T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 0.445T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.445T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.445T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.80T + T^{2} \) |
| 89 | \( 1 + 0.445T + T^{2} \) |
| 97 | \( 1 - 1.24T + T^{2} \) |
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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
−10.12388762482708407791913569800, −8.850376648774954655239907074027, −8.184449228517930695055841618778, −6.91885153996476482150467509091, −6.42852133818765122543118229126, −5.61744517721702811130818183843, −4.78289542946121681451821813265, −3.81724272740992495354642968307, −2.92324067789413949648980233704, −1.55848652556559103968225319469,
1.55848652556559103968225319469, 2.92324067789413949648980233704, 3.81724272740992495354642968307, 4.78289542946121681451821813265, 5.61744517721702811130818183843, 6.42852133818765122543118229126, 6.91885153996476482150467509091, 8.184449228517930695055841618778, 8.850376648774954655239907074027, 10.12388762482708407791913569800