| L(s) = 1 | + 1.57·2-s − 1.51·4-s + 5.50·5-s − 13.3·7-s − 8.69·8-s + 9·9-s + 8.68·10-s − 21.1·14-s − 7.64·16-s + 14.1·18-s + 37.0·19-s − 8.34·20-s + 5.33·25-s + 20.2·28-s − 31·31-s + 22.7·32-s − 73.7·35-s − 13.6·36-s + 58.3·38-s − 47.8·40-s − 76.3·41-s + 49.5·45-s − 30·47-s + 130.·49-s + 8.41·50-s + 116.·56-s − 13.3·59-s + ⋯ |
| L(s) = 1 | + 0.788·2-s − 0.378·4-s + 1.10·5-s − 1.91·7-s − 1.08·8-s + 9-s + 0.868·10-s − 1.50·14-s − 0.478·16-s + 0.788·18-s + 1.94·19-s − 0.417·20-s + 0.213·25-s + 0.723·28-s − 31-s + 0.709·32-s − 2.10·35-s − 0.378·36-s + 1.53·38-s − 1.19·40-s − 1.86·41-s + 1.10·45-s − 0.638·47-s + 2.65·49-s + 0.168·50-s + 2.07·56-s − 0.225·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.274064865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.274064865\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + 31T \) |
| good | 2 | \( 1 - 1.57T + 4T^{2} \) |
| 3 | \( 1 - 9T^{2} \) |
| 5 | \( 1 - 5.50T + 25T^{2} \) |
| 7 | \( 1 + 13.3T + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 37.0T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 76.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 + 30T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + 13.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 10T + 4.48e3T^{2} \) |
| 71 | \( 1 - 131.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 89.0T + 9.40e3T^{2} \) |
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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
−16.49583606628692389593941201340, −15.45945222768043041424219501010, −13.81679387811514757617535014104, −13.24259968301571057370083647193, −12.32709830123748623831638736378, −9.892558317412207671577879839023, −9.442905988699880556415678977692, −6.75705249553714539170057581473, −5.48123695636188947637565074060, −3.41587889882453997469164657778,
3.41587889882453997469164657778, 5.48123695636188947637565074060, 6.75705249553714539170057581473, 9.442905988699880556415678977692, 9.892558317412207671577879839023, 12.32709830123748623831638736378, 13.24259968301571057370083647193, 13.81679387811514757617535014104, 15.45945222768043041424219501010, 16.49583606628692389593941201340
