| L(s) = 1 | + 1.61·2-s − 3.23·3-s + 0.618·4-s + 5-s − 5.23·6-s + 0.236·7-s − 2.23·8-s + 7.47·9-s + 1.61·10-s + 2·11-s − 2.00·12-s − 3.23·13-s + 0.381·14-s − 3.23·15-s − 4.85·16-s + 0.763·17-s + 12.0·18-s − 2.23·19-s + 0.618·20-s − 0.763·21-s + 3.23·22-s + 5.70·23-s + 7.23·24-s − 4·25-s − 5.23·26-s − 14.4·27-s + 0.145·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 1.86·3-s + 0.309·4-s + 0.447·5-s − 2.13·6-s + 0.0892·7-s − 0.790·8-s + 2.49·9-s + 0.511·10-s + 0.603·11-s − 0.577·12-s − 0.897·13-s + 0.102·14-s − 0.835·15-s − 1.21·16-s + 0.185·17-s + 2.84·18-s − 0.512·19-s + 0.138·20-s − 0.166·21-s + 0.689·22-s + 1.19·23-s + 1.47·24-s − 0.800·25-s − 1.02·26-s − 2.78·27-s + 0.0275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7461698958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7461698958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 - 8.18T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 9.18T + 71T^{2} \) |
| 73 | \( 1 - 8.47T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−17.15432508999057702779671121261, −15.79402375241501110804903952077, −14.48246076460290596164391958315, −13.01060856769596994934602495263, −12.19411267640808601410552212031, −11.18435457248788706232861087885, −9.694681048392266970713798808449, −6.75174651250368878590349717473, −5.63181617197644712048767615573, −4.52821779104957415480228032986,
4.52821779104957415480228032986, 5.63181617197644712048767615573, 6.75174651250368878590349717473, 9.694681048392266970713798808449, 11.18435457248788706232861087885, 12.19411267640808601410552212031, 13.01060856769596994934602495263, 14.48246076460290596164391958315, 15.79402375241501110804903952077, 17.15432508999057702779671121261
