| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 8.95·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s − 17.9·10-s + 61.7·11-s − 12·12-s − 54.4·13-s − 14·14-s − 26.8·15-s + 16·16-s + 29.1·17-s − 18·18-s − 157.·19-s + 35.8·20-s − 21·21-s − 123.·22-s − 23·23-s + 24·24-s − 44.8·25-s + 108.·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.800·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.566·10-s + 1.69·11-s − 0.288·12-s − 1.16·13-s − 0.267·14-s − 0.462·15-s + 0.250·16-s + 0.416·17-s − 0.235·18-s − 1.89·19-s + 0.400·20-s − 0.218·21-s − 1.19·22-s − 0.208·23-s + 0.204·24-s − 0.358·25-s + 0.821·26-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 | \( 1 - 8.95T + 125T^{2} \) |
| 11 | \( 1 - 61.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 54.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 29.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 157.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 25.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 254.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 101.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 155.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 476.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 634.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 547.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 174.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 53.4T + 2.26e5T^{2} \) |
| 67 | \( 1 - 593.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 416.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 378.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 423.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 71.9T + 5.71e5T^{2} \) |
| 89 | \( 1 - 15.3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.74e3T + 9.12e5T^{2} \) |
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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
−9.446905008545849682726468241054, −8.521398839981934670429970492245, −7.49726058347119884345276788817, −6.56472528139736654846642056867, −6.04827390074006714954859960722, −4.91687442536395843365182062838, −3.86894129587957972531305030970, −2.20713957933955012412737499539, −1.45065836181173207222528276035, 0,
1.45065836181173207222528276035, 2.20713957933955012412737499539, 3.86894129587957972531305030970, 4.91687442536395843365182062838, 6.04827390074006714954859960722, 6.56472528139736654846642056867, 7.49726058347119884345276788817, 8.521398839981934670429970492245, 9.446905008545849682726468241054
