L(s) = 1 | − 0.390·2-s + 3.60·3-s − 7.84·4-s − 7.52·5-s − 1.40·6-s + 19.5·7-s + 6.18·8-s − 13.9·9-s + 2.93·10-s + 45.8·11-s − 28.3·12-s − 7.62·14-s − 27.1·15-s + 60.3·16-s + 86.5·17-s + 5.45·18-s + 148.·19-s + 59.0·20-s + 70.5·21-s − 17.8·22-s − 91.5·23-s + 22.3·24-s − 68.4·25-s − 147.·27-s − 153.·28-s + 258.·29-s + 10.5·30-s + ⋯ |
L(s) = 1 | − 0.137·2-s + 0.694·3-s − 0.980·4-s − 0.672·5-s − 0.0958·6-s + 1.05·7-s + 0.273·8-s − 0.517·9-s + 0.0927·10-s + 1.25·11-s − 0.681·12-s − 0.145·14-s − 0.467·15-s + 0.943·16-s + 1.23·17-s + 0.0713·18-s + 1.79·19-s + 0.659·20-s + 0.733·21-s − 0.173·22-s − 0.829·23-s + 0.189·24-s − 0.547·25-s − 1.05·27-s − 1.03·28-s + 1.65·29-s + 0.0644·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.663186260\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.663186260\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + 0.390T + 8T^{2} \) |
| 3 | \( 1 - 3.60T + 27T^{2} \) |
| 5 | \( 1 + 7.52T + 125T^{2} \) |
| 7 | \( 1 - 19.5T + 343T^{2} \) |
| 11 | \( 1 - 45.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 86.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 148.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 91.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 258.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 31.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 148.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 95.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 80.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 94.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 493.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 575.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 40.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 601.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 518.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.05e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 320.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 32.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 450.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 231.T + 9.12e5T^{2} \) |
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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
−12.01712164814778279426859421254, −11.66297656465704198516348012154, −10.04213958262747740338664794806, −9.121077306803018730179518342169, −8.184761353818233308126072410603, −7.63984177622862917323455221424, −5.66234074927463592786155070105, −4.38274930129402736804609056663, −3.33218043576476035395280837059, −1.12804639511102488251946898927,
1.12804639511102488251946898927, 3.33218043576476035395280837059, 4.38274930129402736804609056663, 5.66234074927463592786155070105, 7.63984177622862917323455221424, 8.184761353818233308126072410603, 9.121077306803018730179518342169, 10.04213958262747740338664794806, 11.66297656465704198516348012154, 12.01712164814778279426859421254