| L(s) = 1 | − 5-s − 5·11-s − 13-s − 7·17-s + 6·19-s + 3·23-s + 25-s − 6·29-s + 4·31-s + 7·37-s − 3·41-s + 10·47-s − 6·53-s + 5·55-s + 9·59-s − 2·61-s + 65-s − 9·67-s + 16·71-s − 73-s + 3·79-s − 16·83-s + 7·85-s − 5·89-s − 6·95-s + 17·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.50·11-s − 0.277·13-s − 1.69·17-s + 1.37·19-s + 0.625·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 1.15·37-s − 0.468·41-s + 1.45·47-s − 0.824·53-s + 0.674·55-s + 1.17·59-s − 0.256·61-s + 0.124·65-s − 1.09·67-s + 1.89·71-s − 0.117·73-s + 0.337·79-s − 1.75·83-s + 0.759·85-s − 0.529·89-s − 0.615·95-s + 1.72·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.290688629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.290688629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.92628463363925, −12.63924712388265, −11.88759541976361, −11.53379524960283, −10.99675700805495, −10.81288039614097, −10.11238004150533, −9.687539906152472, −9.189268648388718, −8.693140178437397, −8.155856479568762, −7.732852856915023, −7.204130098362933, −6.975980695491218, −6.170903599648502, −5.663623899823959, −5.153307476661520, −4.678787401818348, −4.252972494276364, −3.512902356600709, −2.947498191814915, −2.477377184402218, −1.963376288477511, −0.9902328355465076, −0.3566565163938399,
0.3566565163938399, 0.9902328355465076, 1.963376288477511, 2.477377184402218, 2.947498191814915, 3.512902356600709, 4.252972494276364, 4.678787401818348, 5.153307476661520, 5.663623899823959, 6.170903599648502, 6.975980695491218, 7.204130098362933, 7.732852856915023, 8.155856479568762, 8.693140178437397, 9.189268648388718, 9.687539906152472, 10.11238004150533, 10.81288039614097, 10.99675700805495, 11.53379524960283, 11.88759541976361, 12.63924712388265, 12.92628463363925
