| L(s) = 1 | − 1.58·2-s − 3.01·3-s + 0.515·4-s + 5-s + 4.78·6-s + 7-s + 2.35·8-s + 6.11·9-s − 1.58·10-s − 1.55·12-s + 3.17·13-s − 1.58·14-s − 3.01·15-s − 4.76·16-s − 6.29·17-s − 9.69·18-s + 1.15·19-s + 0.515·20-s − 3.01·21-s + 5.96·23-s − 7.10·24-s + 25-s − 5.02·26-s − 9.39·27-s + 0.515·28-s + 8.32·29-s + 4.78·30-s + ⋯ |
| L(s) = 1 | − 1.12·2-s − 1.74·3-s + 0.257·4-s + 0.447·5-s + 1.95·6-s + 0.377·7-s + 0.832·8-s + 2.03·9-s − 0.501·10-s − 0.449·12-s + 0.879·13-s − 0.423·14-s − 0.779·15-s − 1.19·16-s − 1.52·17-s − 2.28·18-s + 0.264·19-s + 0.115·20-s − 0.658·21-s + 1.24·23-s − 1.45·24-s + 0.200·25-s − 0.986·26-s − 1.80·27-s + 0.0973·28-s + 1.54·29-s + 0.874·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6823318182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6823318182\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.58T + 2T^{2} \) |
| 3 | \( 1 + 3.01T + 3T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 + 6.29T + 17T^{2} \) |
| 19 | \( 1 - 1.15T + 19T^{2} \) |
| 23 | \( 1 - 5.96T + 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 - 8.69T + 31T^{2} \) |
| 37 | \( 1 - 3.37T + 37T^{2} \) |
| 41 | \( 1 - 6.78T + 41T^{2} \) |
| 43 | \( 1 - 5.11T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 8.18T + 53T^{2} \) |
| 59 | \( 1 - 7.10T + 59T^{2} \) |
| 61 | \( 1 - 7.27T + 61T^{2} \) |
| 67 | \( 1 + 8.71T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 4.48T + 73T^{2} \) |
| 79 | \( 1 + 3.30T + 79T^{2} \) |
| 83 | \( 1 + 7.14T + 83T^{2} \) |
| 89 | \( 1 - 0.654T + 89T^{2} \) |
| 97 | \( 1 - 8.73T + 97T^{2} \) |
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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
−8.609787407559088642566211319405, −7.62778646432485130136490585395, −6.83316099185965961168742727226, −6.34485158108969758945195671646, −5.57225523082864911187357370971, −4.60122894529589181855290436505, −4.37520578372013478891082013567, −2.54210120951362426752110705515, −1.25674030407372232291858923474, −0.73597463764657384154205215424,
0.73597463764657384154205215424, 1.25674030407372232291858923474, 2.54210120951362426752110705515, 4.37520578372013478891082013567, 4.60122894529589181855290436505, 5.57225523082864911187357370971, 6.34485158108969758945195671646, 6.83316099185965961168742727226, 7.62778646432485130136490585395, 8.609787407559088642566211319405
