| L(s) = 1 | + 8.36·2-s + 38·4-s − 45·7-s + 50.1·8-s − 334.·11-s − 1.04e3·13-s − 376.·14-s − 796·16-s − 1.27e3·17-s + 1.15e3·19-s − 2.80e3·22-s + 3.81e3·23-s − 8.74e3·26-s − 1.71e3·28-s − 3.68e3·29-s + 3.63e3·31-s − 8.26e3·32-s − 1.06e4·34-s − 3.11e3·37-s + 9.69e3·38-s − 1.77e4·41-s + 1.03e4·43-s − 1.27e4·44-s + 3.19e4·46-s − 1.35e4·47-s − 1.47e4·49-s − 3.97e4·52-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.18·4-s − 0.347·7-s + 0.277·8-s − 0.833·11-s − 1.71·13-s − 0.513·14-s − 0.777·16-s − 1.06·17-s + 0.736·19-s − 1.23·22-s + 1.50·23-s − 2.53·26-s − 0.412·28-s − 0.812·29-s + 0.678·31-s − 1.42·32-s − 1.57·34-s − 0.373·37-s + 1.08·38-s − 1.64·41-s + 0.854·43-s − 0.990·44-s + 2.22·46-s − 0.892·47-s − 0.879·49-s − 2.03·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 8.36T + 32T^{2} \) |
| 7 | \( 1 + 45T + 1.68e4T^{2} \) |
| 11 | \( 1 + 334.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.27e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.15e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.81e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.68e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.63e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.11e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.77e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.97e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.87e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.61e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.06e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.47e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.83e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.57e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.20e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.17e4T + 8.58e9T^{2} \) |
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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
−11.26193556937044919801696363946, −10.07229732276547028347304253400, −9.004118216724519847416156084935, −7.45314690111052363913801440029, −6.62211747726913765869515085152, −5.26625775222954581518111285767, −4.73079121027576404054921797681, −3.26265524999893725840030717542, −2.35015981367577130817835094022, 0,
2.35015981367577130817835094022, 3.26265524999893725840030717542, 4.73079121027576404054921797681, 5.26625775222954581518111285767, 6.62211747726913765869515085152, 7.45314690111052363913801440029, 9.004118216724519847416156084935, 10.07229732276547028347304253400, 11.26193556937044919801696363946
