L(s) = 1 | − 3-s − 5-s + 9-s − 11-s + 6·13-s + 15-s + 2·17-s − 2·19-s + 4·23-s + 25-s − 27-s − 6·29-s + 33-s − 2·37-s − 6·39-s − 4·41-s + 4·43-s − 45-s − 2·51-s + 4·53-s + 55-s + 2·57-s − 6·59-s + 2·61-s − 6·65-s + 14·67-s − 4·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s + 1.66·13-s + 0.258·15-s + 0.485·17-s − 0.458·19-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.174·33-s − 0.328·37-s − 0.960·39-s − 0.624·41-s + 0.609·43-s − 0.149·45-s − 0.280·51-s + 0.549·53-s + 0.134·55-s + 0.264·57-s − 0.781·59-s + 0.256·61-s − 0.744·65-s + 1.71·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.42403048708944, −14.87330115265681, −14.30179438562677, −13.57277072792905, −13.14529827890915, −12.73330899968775, −12.08037558486932, −11.50257449022604, −11.06525615427727, −10.66364803260315, −10.12274473244587, −9.332852347843729, −8.788286051955066, −8.292963078328418, −7.645677621180102, −7.065334943110508, −6.472210053722106, −5.839503224193533, −5.387849287913223, −4.663132575852891, −3.905110500030437, −3.517740925470902, −2.671970592263774, −1.639666276901872, −0.9981093122521042, 0,
0.9981093122521042, 1.639666276901872, 2.671970592263774, 3.517740925470902, 3.905110500030437, 4.663132575852891, 5.387849287913223, 5.839503224193533, 6.472210053722106, 7.065334943110508, 7.645677621180102, 8.292963078328418, 8.788286051955066, 9.332852347843729, 10.12274473244587, 10.66364803260315, 11.06525615427727, 11.50257449022604, 12.08037558486932, 12.73330899968775, 13.14529827890915, 13.57277072792905, 14.30179438562677, 14.87330115265681, 15.42403048708944