| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 6.51·5-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s + 13.0·10-s − 52.6·11-s + 12·12-s − 71.8·13-s − 14·14-s + 19.5·15-s + 16·16-s − 54.7·17-s + 18·18-s + 17.2·19-s + 26.0·20-s − 21·21-s − 105.·22-s + 23·23-s + 24·24-s − 82.5·25-s − 143.·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.582·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.411·10-s − 1.44·11-s + 0.288·12-s − 1.53·13-s − 0.267·14-s + 0.336·15-s + 0.250·16-s − 0.781·17-s + 0.235·18-s + 0.208·19-s + 0.291·20-s − 0.218·21-s − 1.02·22-s + 0.208·23-s + 0.204·24-s − 0.660·25-s − 1.08·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 6.51T + 125T^{2} \) |
| 11 | \( 1 + 52.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 71.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 54.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 17.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 24.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 261.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 409.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 58.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 127.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 144.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 507.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 56.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 416.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 382.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 772.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 386.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 752.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 385.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 94.1T + 9.12e5T^{2} \) |
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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
−9.388975777977523053880970396037, −8.276477406211608595859300707415, −7.39772813597965606968224007239, −6.72772109310596203297178928554, −5.46387278121486052556011414698, −4.97223202093162102469818764455, −3.71469869486425790949563681097, −2.61678938038260337198047103715, −2.05115902440111229499056615689, 0,
2.05115902440111229499056615689, 2.61678938038260337198047103715, 3.71469869486425790949563681097, 4.97223202093162102469818764455, 5.46387278121486052556011414698, 6.72772109310596203297178928554, 7.39772813597965606968224007239, 8.276477406211608595859300707415, 9.388975777977523053880970396037
