| L(s) = 1 | − 0.527·2-s + 1.24·3-s − 1.72·4-s − 0.656·6-s + 3.22·7-s + 1.96·8-s − 1.45·9-s + 11-s − 2.14·12-s − 4.87·13-s − 1.69·14-s + 2.41·16-s + 7.11·17-s + 0.764·18-s + 19-s + 4.01·21-s − 0.527·22-s + 1.12·23-s + 2.44·24-s + 2.56·26-s − 5.53·27-s − 5.54·28-s + 8.47·29-s − 0.926·31-s − 5.19·32-s + 1.24·33-s − 3.75·34-s + ⋯ |
| L(s) = 1 | − 0.372·2-s + 0.718·3-s − 0.861·4-s − 0.267·6-s + 1.21·7-s + 0.693·8-s − 0.483·9-s + 0.301·11-s − 0.618·12-s − 1.35·13-s − 0.453·14-s + 0.602·16-s + 1.72·17-s + 0.180·18-s + 0.229·19-s + 0.875·21-s − 0.112·22-s + 0.234·23-s + 0.498·24-s + 0.503·26-s − 1.06·27-s − 1.04·28-s + 1.57·29-s − 0.166·31-s − 0.918·32-s + 0.216·33-s − 0.643·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.870435938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.870435938\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.527T + 2T^{2} \) |
| 3 | \( 1 - 1.24T + 3T^{2} \) |
| 7 | \( 1 - 3.22T + 7T^{2} \) |
| 13 | \( 1 + 4.87T + 13T^{2} \) |
| 17 | \( 1 - 7.11T + 17T^{2} \) |
| 23 | \( 1 - 1.12T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 0.926T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 7.49T + 41T^{2} \) |
| 43 | \( 1 + 7.22T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 7.92T + 53T^{2} \) |
| 59 | \( 1 - 1.91T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 0.979T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 3.05T + 79T^{2} \) |
| 83 | \( 1 + 2.60T + 83T^{2} \) |
| 89 | \( 1 - 5.38T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.338665383136570472608961200485, −7.60202564589714069484662642263, −7.28130744178078441068854334425, −5.81452308333526169235393053542, −5.15434450573115415249866931915, −4.62720023883456843208842850415, −3.66184503182185092303404418753, −2.84540344504849066589766071853, −1.80492518997956083140819796008, −0.791238087005524469002005564743,
0.791238087005524469002005564743, 1.80492518997956083140819796008, 2.84540344504849066589766071853, 3.66184503182185092303404418753, 4.62720023883456843208842850415, 5.15434450573115415249866931915, 5.81452308333526169235393053542, 7.28130744178078441068854334425, 7.60202564589714069484662642263, 8.338665383136570472608961200485
