| L(s) = 1 | + 0.149·2-s − 6.48·3-s − 7.97·4-s − 10.2·5-s − 0.967·6-s − 29.6·7-s − 2.38·8-s + 15.1·9-s − 1.52·10-s + 38.1·11-s + 51.7·12-s − 4.42·14-s + 66.5·15-s + 63.4·16-s − 71.3·17-s + 2.25·18-s + 10.0·19-s + 81.7·20-s + 192.·21-s + 5.67·22-s − 198.·23-s + 15.4·24-s − 19.8·25-s + 77.2·27-s + 236.·28-s + 30.8·29-s + 9.91·30-s + ⋯ |
| L(s) = 1 | + 0.0527·2-s − 1.24·3-s − 0.997·4-s − 0.917·5-s − 0.0658·6-s − 1.60·7-s − 0.105·8-s + 0.559·9-s − 0.0483·10-s + 1.04·11-s + 1.24·12-s − 0.0844·14-s + 1.14·15-s + 0.991·16-s − 1.01·17-s + 0.0294·18-s + 0.121·19-s + 0.914·20-s + 2.00·21-s + 0.0550·22-s − 1.80·23-s + 0.131·24-s − 0.159·25-s + 0.550·27-s + 1.59·28-s + 0.197·29-s + 0.0603·30-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
| \(L(2)\) |
\(\approx\) |
\(0.2472203315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2472203315\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 0.149T + 8T^{2} \) |
| 3 | \( 1 + 6.48T + 27T^{2} \) |
| 5 | \( 1 + 10.2T + 125T^{2} \) |
| 7 | \( 1 + 29.6T + 343T^{2} \) |
| 11 | \( 1 - 38.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 71.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 10.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 198.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 30.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 151.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 207.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 303.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 12.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 250.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 390.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 156.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 303.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 913.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 249.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 147.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 946.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 417.T + 9.12e5T^{2} \) |
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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
−12.14381828678692321030561155666, −11.64804110584125613923486041464, −10.26438992571910180530716591637, −9.450856241067548057141326927739, −8.280484202553310454565714593446, −6.69310327957016742682501933613, −5.99241782699965280520023346465, −4.50306462490111800151242182330, −3.59443446369436073759021767031, −0.40080922869567823225549715500,
0.40080922869567823225549715500, 3.59443446369436073759021767031, 4.50306462490111800151242182330, 5.99241782699965280520023346465, 6.69310327957016742682501933613, 8.280484202553310454565714593446, 9.450856241067548057141326927739, 10.26438992571910180530716591637, 11.64804110584125613923486041464, 12.14381828678692321030561155666
