| L(s) = 1 | + i·3-s + 4.32i·7-s − 9-s + 0.584·11-s − 0.966i·13-s − 2.32i·17-s + 5.37·19-s − 4.32·21-s + 1.35i·23-s − i·27-s + 0.706·29-s + 8.48·31-s + 0.584i·33-s − 6.11i·37-s + 0.966·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.63i·7-s − 0.333·9-s + 0.176·11-s − 0.267i·13-s − 0.563i·17-s + 1.23·19-s − 0.943·21-s + 0.283i·23-s − 0.192i·27-s + 0.131·29-s + 1.52·31-s + 0.101i·33-s − 1.00i·37-s + 0.154·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.183820378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.183820378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.32iT - 7T^{2} \) |
| 11 | \( 1 - 0.584T + 11T^{2} \) |
| 13 | \( 1 + 0.966iT - 13T^{2} \) |
| 17 | \( 1 + 2.32iT - 17T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 - 1.35iT - 23T^{2} \) |
| 29 | \( 1 - 0.706T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 6.11iT - 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 7.03iT - 43T^{2} \) |
| 47 | \( 1 - 9.06iT - 47T^{2} \) |
| 53 | \( 1 + 8.90iT - 53T^{2} \) |
| 59 | \( 1 - 7.29T + 59T^{2} \) |
| 61 | \( 1 + 4.81T + 61T^{2} \) |
| 67 | \( 1 + 0.376iT - 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 0.213iT - 73T^{2} \) |
| 79 | \( 1 - 7.02T + 79T^{2} \) |
| 83 | \( 1 + 16.1iT - 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 15.7iT - 97T^{2} \) |
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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
−8.049308402068454943510976589524, −7.58035961150106384612247482881, −6.41854780709316541580949436405, −5.93535283432295970809095999248, −5.19289315387326369796452965536, −4.73093846478720202240333836623, −3.64049413016252497698326071210, −2.85092807343592010260537220455, −2.30648824782148390216202906166, −0.937826984756172932020983790222,
0.69873345862886988470925858699, 1.27647094619037519941749876668, 2.43911479996359366600038126516, 3.41858722442667048560565492714, 4.08793342584036234041788479078, 4.79787357395922783079307522282, 5.71676332919930198103597883524, 6.58974333114031146387226417746, 6.96067717449120027633286573976, 7.71821213700516025406518984613
