| L(s) = 1 | − 2-s − 0.709·3-s + 4-s − 2.35·5-s + 0.709·6-s + 2.33·7-s − 8-s − 2.49·9-s + 2.35·10-s − 11-s − 0.709·12-s + 0.905·13-s − 2.33·14-s + 1.66·15-s + 16-s − 2.18·17-s + 2.49·18-s − 2.35·20-s − 1.65·21-s + 22-s + 5.05·23-s + 0.709·24-s + 0.531·25-s − 0.905·26-s + 3.89·27-s + 2.33·28-s + 1.38·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.409·3-s + 0.5·4-s − 1.05·5-s + 0.289·6-s + 0.881·7-s − 0.353·8-s − 0.832·9-s + 0.743·10-s − 0.301·11-s − 0.204·12-s + 0.251·13-s − 0.623·14-s + 0.430·15-s + 0.250·16-s − 0.529·17-s + 0.588·18-s − 0.525·20-s − 0.360·21-s + 0.213·22-s + 1.05·23-s + 0.144·24-s + 0.106·25-s − 0.177·26-s + 0.750·27-s + 0.440·28-s + 0.257·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.709T + 3T^{2} \) |
| 5 | \( 1 + 2.35T + 5T^{2} \) |
| 7 | \( 1 - 2.33T + 7T^{2} \) |
| 13 | \( 1 - 0.905T + 13T^{2} \) |
| 17 | \( 1 + 2.18T + 17T^{2} \) |
| 23 | \( 1 - 5.05T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 - 3.97T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 3.04T + 47T^{2} \) |
| 53 | \( 1 - 6.64T + 53T^{2} \) |
| 59 | \( 1 + 9.41T + 59T^{2} \) |
| 61 | \( 1 - 8.38T + 61T^{2} \) |
| 67 | \( 1 - 5.56T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 9.09T + 73T^{2} \) |
| 79 | \( 1 - 2.93T + 79T^{2} \) |
| 83 | \( 1 - 9.40T + 83T^{2} \) |
| 89 | \( 1 - 7.21T + 89T^{2} \) |
| 97 | \( 1 - 6.40T + 97T^{2} \) |
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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
−7.70504647807385550515634418549, −6.95056331479083620417340447254, −6.23966775549184584781627209897, −5.34771870276265096416664095782, −4.79945378787813364987260811516, −3.87329677034843074325847419701, −3.03918465400177794297332619299, −2.12382070443963560668763591807, −0.976687093388699095865992117889, 0,
0.976687093388699095865992117889, 2.12382070443963560668763591807, 3.03918465400177794297332619299, 3.87329677034843074325847419701, 4.79945378787813364987260811516, 5.34771870276265096416664095782, 6.23966775549184584781627209897, 6.95056331479083620417340447254, 7.70504647807385550515634418549
