| L(s) = 1 | − 0.445·3-s + 4-s + 1.80·5-s − 0.801·9-s − 11-s − 0.445·12-s − 0.801·15-s + 16-s + 1.80·20-s + 1.24·23-s + 2.24·25-s + 0.801·27-s − 1.24·31-s + 0.445·33-s − 0.801·36-s + 0.445·37-s − 44-s − 1.44·45-s − 1.24·47-s − 0.445·48-s + 49-s − 1.80·53-s − 1.80·55-s + 0.445·59-s − 0.801·60-s + 64-s − 2·67-s + ⋯ |
| L(s) = 1 | − 0.445·3-s + 4-s + 1.80·5-s − 0.801·9-s − 11-s − 0.445·12-s − 0.801·15-s + 16-s + 1.80·20-s + 1.24·23-s + 2.24·25-s + 0.801·27-s − 1.24·31-s + 0.445·33-s − 0.801·36-s + 0.445·37-s − 44-s − 1.44·45-s − 1.24·47-s − 0.445·48-s + 49-s − 1.80·53-s − 1.80·55-s + 0.445·59-s − 0.801·60-s + 64-s − 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.553571823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.553571823\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 + 0.445T + T^{2} \) |
| 5 | \( 1 - 1.80T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.24T + T^{2} \) |
| 37 | \( 1 - 0.445T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.24T + T^{2} \) |
| 53 | \( 1 + 1.80T + T^{2} \) |
| 59 | \( 1 - 0.445T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 2T + T^{2} \) |
| 71 | \( 1 - 0.445T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.24T + T^{2} \) |
| 97 | \( 1 - 1.80T + T^{2} \) |
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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
−9.523467115314262790312089471736, −8.779192674310314910422654952488, −7.74446943293197462566540858947, −6.82181685127869731239612250549, −6.15388869568747218067232963669, −5.52728993749984777505149315653, −5.01268512267147233020904550420, −3.09392753285088635638293147180, −2.49959579678107293328132698038, −1.49616530311272384194224493215,
1.49616530311272384194224493215, 2.49959579678107293328132698038, 3.09392753285088635638293147180, 5.01268512267147233020904550420, 5.52728993749984777505149315653, 6.15388869568747218067232963669, 6.82181685127869731239612250549, 7.74446943293197462566540858947, 8.779192674310314910422654952488, 9.523467115314262790312089471736
