| L(s) = 1 | − 2.79·2-s − 3-s + 5.79·4-s + 3·5-s + 2.79·6-s + 7-s − 10.5·8-s + 9-s − 8.37·10-s − 11-s − 5.79·12-s + 13-s − 2.79·14-s − 3·15-s + 17.9·16-s − 1.58·17-s − 2.79·18-s + 2.58·19-s + 17.3·20-s − 21-s + 2.79·22-s + 3.58·23-s + 10.5·24-s + 4·25-s − 2.79·26-s − 27-s + 5.79·28-s + ⋯ |
| L(s) = 1 | − 1.97·2-s − 0.577·3-s + 2.89·4-s + 1.34·5-s + 1.13·6-s + 0.377·7-s − 3.74·8-s + 0.333·9-s − 2.64·10-s − 0.301·11-s − 1.67·12-s + 0.277·13-s − 0.746·14-s − 0.774·15-s + 4.48·16-s − 0.383·17-s − 0.657·18-s + 0.592·19-s + 3.88·20-s − 0.218·21-s + 0.595·22-s + 0.747·23-s + 2.16·24-s + 0.800·25-s − 0.547·26-s − 0.192·27-s + 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5835294592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5835294592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 1.58T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 - 3.58T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 7.16T + 41T^{2} \) |
| 43 | \( 1 + 7.58T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 0.417T + 53T^{2} \) |
| 59 | \( 1 + 4.58T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 0.582T + 67T^{2} \) |
| 71 | \( 1 + 7.16T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 2.41T + 83T^{2} \) |
| 89 | \( 1 + 9.16T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−11.70276715553557933844493548601, −10.82866716274895845714783537674, −10.18835345295472751493961142304, −9.364021806975089841297799209259, −8.524475969918972472559858011629, −7.30743486816592193257801011398, −6.38517709956669484565179440345, −5.45725405618762075598023319108, −2.56982956210564980010893424414, −1.25745751196044836850926097415,
1.25745751196044836850926097415, 2.56982956210564980010893424414, 5.45725405618762075598023319108, 6.38517709956669484565179440345, 7.30743486816592193257801011398, 8.524475969918972472559858011629, 9.364021806975089841297799209259, 10.18835345295472751493961142304, 10.82866716274895845714783537674, 11.70276715553557933844493548601
