L(s) = 1 | + 2.19·2-s − 2.27·3-s + 2.80·4-s − 4.99·6-s − 1.38·7-s + 1.75·8-s + 2.19·9-s − 4.99·11-s − 6.38·12-s − 2.43·13-s − 3.04·14-s − 1.75·16-s − 4.71·17-s + 4.80·18-s + 5.74·19-s + 3.16·21-s − 10.9·22-s + 1.38·23-s − 3.99·24-s − 5.33·26-s + 1.84·27-s − 3.89·28-s − 5.16·29-s − 31-s − 7.35·32-s + 11.3·33-s − 10.3·34-s + ⋯ |
L(s) = 1 | + 1.54·2-s − 1.31·3-s + 1.40·4-s − 2.03·6-s − 0.525·7-s + 0.620·8-s + 0.730·9-s − 1.50·11-s − 1.84·12-s − 0.675·13-s − 0.813·14-s − 0.438·16-s − 1.14·17-s + 1.13·18-s + 1.31·19-s + 0.691·21-s − 2.33·22-s + 0.289·23-s − 0.816·24-s − 1.04·26-s + 0.354·27-s − 0.735·28-s − 0.959·29-s − 0.179·31-s − 1.30·32-s + 1.98·33-s − 1.77·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 - 2.19T + 2T^{2} \) |
| 3 | \( 1 + 2.27T + 3T^{2} \) |
| 7 | \( 1 + 1.38T + 7T^{2} \) |
| 11 | \( 1 + 4.99T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 + 4.71T + 17T^{2} \) |
| 19 | \( 1 - 5.74T + 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 + 5.16T + 29T^{2} \) |
| 37 | \( 1 - 4.31T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 2.15T + 43T^{2} \) |
| 47 | \( 1 - 8.20T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 0.672T + 59T^{2} \) |
| 61 | \( 1 - 1.44T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 2.80T + 71T^{2} \) |
| 73 | \( 1 + 2.85T + 73T^{2} \) |
| 79 | \( 1 - 6.85T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 8.15T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−10.27699085661625832425720126960, −9.260876176320719473397778302939, −7.71356823351601381409413519694, −6.80439540217213358486665127649, −6.06984368134385137104138961596, −5.09037362219977112743454119641, −4.95584556129276486877954607778, −3.49178735540773958891886497528, −2.43716029150059304967424314359, 0,
2.43716029150059304967424314359, 3.49178735540773958891886497528, 4.95584556129276486877954607778, 5.09037362219977112743454119641, 6.06984368134385137104138961596, 6.80439540217213358486665127649, 7.71356823351601381409413519694, 9.260876176320719473397778302939, 10.27699085661625832425720126960