L(s) = 1 | − 2-s + 3.39·3-s + 4-s − 3.97·5-s − 3.39·6-s + 1.24·7-s − 8-s + 8.55·9-s + 3.97·10-s + 2.00·11-s + 3.39·12-s + 0.921·13-s − 1.24·14-s − 13.5·15-s + 16-s − 1.13·17-s − 8.55·18-s + 4.11·19-s − 3.97·20-s + 4.24·21-s − 2.00·22-s − 23-s − 3.39·24-s + 10.8·25-s − 0.921·26-s + 18.8·27-s + 1.24·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.96·3-s + 0.5·4-s − 1.77·5-s − 1.38·6-s + 0.471·7-s − 0.353·8-s + 2.85·9-s + 1.25·10-s + 0.603·11-s + 0.981·12-s + 0.255·13-s − 0.333·14-s − 3.49·15-s + 0.250·16-s − 0.276·17-s − 2.01·18-s + 0.944·19-s − 0.889·20-s + 0.926·21-s − 0.426·22-s − 0.208·23-s − 0.693·24-s + 2.16·25-s − 0.180·26-s + 3.63·27-s + 0.235·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.638566447\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.638566447\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 - 3.39T + 3T^{2} \) |
| 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 - 1.24T + 7T^{2} \) |
| 11 | \( 1 - 2.00T + 11T^{2} \) |
| 13 | \( 1 - 0.921T + 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 - 4.11T + 19T^{2} \) |
| 29 | \( 1 + 9.08T + 29T^{2} \) |
| 31 | \( 1 - 2.81T + 31T^{2} \) |
| 37 | \( 1 - 5.84T + 37T^{2} \) |
| 41 | \( 1 - 6.93T + 41T^{2} \) |
| 43 | \( 1 - 9.14T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 7.66T + 53T^{2} \) |
| 59 | \( 1 - 0.834T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 + 4.14T + 67T^{2} \) |
| 71 | \( 1 - 1.50T + 71T^{2} \) |
| 73 | \( 1 + 7.72T + 73T^{2} \) |
| 79 | \( 1 + 4.63T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−7.989677453157387541886802345549, −7.66845958187290054391414390360, −7.28112865445666605647835468464, −6.35549609116155817435833792581, −4.78590307374516067318349412150, −4.08569904163618259458349287964, −3.53938240077874691588447803003, −2.88715119801482422510527398638, −1.85450244038774316340315016012, −0.903758387358936716617767899075,
0.903758387358936716617767899075, 1.85450244038774316340315016012, 2.88715119801482422510527398638, 3.53938240077874691588447803003, 4.08569904163618259458349287964, 4.78590307374516067318349412150, 6.35549609116155817435833792581, 7.28112865445666605647835468464, 7.66845958187290054391414390360, 7.989677453157387541886802345549