L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 3·11-s + 12-s + 1.41·13-s − 15-s + 16-s − 1.41·17-s + 18-s − 3.41·19-s − 20-s − 3·22-s − 23-s + 24-s − 4·25-s + 1.41·26-s + 27-s − 0.757·29-s − 30-s + 4.65·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.904·11-s + 0.288·12-s + 0.392·13-s − 0.258·15-s + 0.250·16-s − 0.342·17-s + 0.235·18-s − 0.783·19-s − 0.223·20-s − 0.639·22-s − 0.208·23-s + 0.204·24-s − 0.800·25-s + 0.277·26-s + 0.192·27-s − 0.140·29-s − 0.182·30-s + 0.836·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 29 | \( 1 + 0.757T + 29T^{2} \) |
| 31 | \( 1 - 4.65T + 31T^{2} \) |
| 37 | \( 1 + 9.07T + 37T^{2} \) |
| 41 | \( 1 - 1.75T + 41T^{2} \) |
| 43 | \( 1 + 7.65T + 43T^{2} \) |
| 47 | \( 1 + 9.31T + 47T^{2} \) |
| 53 | \( 1 + 2.17T + 53T^{2} \) |
| 59 | \( 1 + 0.757T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 4.34T + 67T^{2} \) |
| 71 | \( 1 - 5.89T + 71T^{2} \) |
| 73 | \( 1 - 5.65T + 73T^{2} \) |
| 79 | \( 1 + 0.414T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 3.58T + 97T^{2} \) |
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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
−7.83421428360826378941574040302, −6.79101174619404855877967121766, −6.32850783028275648744787830735, −5.32777969005790222242473949296, −4.71294388086849160000394949267, −3.90520579183953337706547867655, −3.30027518556795537476864734477, −2.43836830321349846841350602451, −1.63480108677527536968796946855, 0,
1.63480108677527536968796946855, 2.43836830321349846841350602451, 3.30027518556795537476864734477, 3.90520579183953337706547867655, 4.71294388086849160000394949267, 5.32777969005790222242473949296, 6.32850783028275648744787830735, 6.79101174619404855877967121766, 7.83421428360826378941574040302