L(s) = 1 | − 5·5-s + 24.7·7-s − 8.16·11-s − 46.7·13-s + 60.3·17-s − 111.·19-s + 36.9·23-s + 25·25-s + 33.2·29-s − 124.·31-s − 123.·35-s − 438.·37-s + 508.·41-s − 48.5·43-s − 248.·47-s + 271.·49-s − 320.·53-s + 40.8·55-s − 652.·59-s + 693.·61-s + 233.·65-s − 12.0·67-s + 1.16e3·71-s − 122.·73-s − 202.·77-s − 441.·79-s + 428.·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.33·7-s − 0.223·11-s − 0.997·13-s + 0.860·17-s − 1.34·19-s + 0.334·23-s + 0.200·25-s + 0.212·29-s − 0.720·31-s − 0.598·35-s − 1.94·37-s + 1.93·41-s − 0.172·43-s − 0.769·47-s + 0.791·49-s − 0.830·53-s + 0.100·55-s − 1.43·59-s + 1.45·61-s + 0.446·65-s − 0.0219·67-s + 1.95·71-s − 0.195·73-s − 0.299·77-s − 0.629·79-s + 0.566·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
good | 7 | \( 1 - 24.7T + 343T^{2} \) |
| 11 | \( 1 + 8.16T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 60.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 36.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 33.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 438.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 508.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 48.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 248.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 320.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 652.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 693.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 12.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 122.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 441.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 428.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.54e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 500.T + 9.12e5T^{2} \) |
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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
−8.958302060001968415123582014030, −8.131224180690229464624917330075, −7.61014114423984114972701984111, −6.69527127558629306174959252850, −5.40826933796911370235317525215, −4.79787542421900943883923591816, −3.85424453797724763450498610233, −2.54203006092817928821594324137, −1.47578136262274374908855908286, 0,
1.47578136262274374908855908286, 2.54203006092817928821594324137, 3.85424453797724763450498610233, 4.79787542421900943883923591816, 5.40826933796911370235317525215, 6.69527127558629306174959252850, 7.61014114423984114972701984111, 8.131224180690229464624917330075, 8.958302060001968415123582014030