L(s) = 1 | − 2.41·2-s + 3-s + 3.82·4-s − 3.41·5-s − 2.41·6-s + 0.828·7-s − 4.41·8-s + 9-s + 8.24·10-s + 11-s + 3.82·12-s − 13-s − 1.99·14-s − 3.41·15-s + 2.99·16-s − 2.58·17-s − 2.41·18-s − 6·19-s − 13.0·20-s + 0.828·21-s − 2.41·22-s + 4.82·23-s − 4.41·24-s + 6.65·25-s + 2.41·26-s + 27-s + 3.17·28-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 0.577·3-s + 1.91·4-s − 1.52·5-s − 0.985·6-s + 0.313·7-s − 1.56·8-s + 0.333·9-s + 2.60·10-s + 0.301·11-s + 1.10·12-s − 0.277·13-s − 0.534·14-s − 0.881·15-s + 0.749·16-s − 0.627·17-s − 0.569·18-s − 1.37·19-s − 2.92·20-s + 0.180·21-s − 0.514·22-s + 1.00·23-s − 0.901·24-s + 1.33·25-s + 0.473·26-s + 0.192·27-s + 0.599·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 9.07T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 + 5.17T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 4.48T + 73T^{2} \) |
| 79 | \( 1 - 5.07T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−10.73462947775384612173659269797, −9.572267351407206440797229255036, −8.625211138176998733528565617552, −8.294380618832128264164396604935, −7.31886910412356544374495004208, −6.73456846493927341009354436701, −4.63610094777111649562187410115, −3.39557814695746474405673163071, −1.85487998435817000029845974424, 0,
1.85487998435817000029845974424, 3.39557814695746474405673163071, 4.63610094777111649562187410115, 6.73456846493927341009354436701, 7.31886910412356544374495004208, 8.294380618832128264164396604935, 8.625211138176998733528565617552, 9.572267351407206440797229255036, 10.73462947775384612173659269797