L(s) = 1 | + 5-s + 7-s − 11-s + 13-s − 5·19-s − 6·23-s + 25-s + 7·31-s + 35-s + 8·37-s − 6·41-s + 7·43-s − 3·47-s − 6·49-s − 55-s − 9·59-s + 5·61-s + 65-s − 5·67-s − 12·71-s − 4·73-s − 77-s − 11·79-s − 9·83-s + 3·89-s + 91-s − 5·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.301·11-s + 0.277·13-s − 1.14·19-s − 1.25·23-s + 1/5·25-s + 1.25·31-s + 0.169·35-s + 1.31·37-s − 0.937·41-s + 1.06·43-s − 0.437·47-s − 6/7·49-s − 0.134·55-s − 1.17·59-s + 0.640·61-s + 0.124·65-s − 0.610·67-s − 1.42·71-s − 0.468·73-s − 0.113·77-s − 1.23·79-s − 0.987·83-s + 0.317·89-s + 0.104·91-s − 0.512·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.911865732\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.911865732\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.60586568351050, −13.30307531618842, −12.84422867536066, −12.23653684786984, −11.78448360190674, −11.29256831238726, −10.70766409794596, −10.30458418762659, −9.859548466133539, −9.306217226905446, −8.712099251490613, −8.151651520866070, −7.941704868581619, −7.184499615556540, −6.530745703946303, −6.108295104228219, −5.698150796546077, −4.981563762375152, −4.253105337799564, −4.186197412843718, −3.045123293476696, −2.677102566736916, −1.870311296594808, −1.425990186214685, −0.4204566796334598,
0.4204566796334598, 1.425990186214685, 1.870311296594808, 2.677102566736916, 3.045123293476696, 4.186197412843718, 4.253105337799564, 4.981563762375152, 5.698150796546077, 6.108295104228219, 6.530745703946303, 7.184499615556540, 7.941704868581619, 8.151651520866070, 8.712099251490613, 9.306217226905446, 9.859548466133539, 10.30458418762659, 10.70766409794596, 11.29256831238726, 11.78448360190674, 12.23653684786984, 12.84422867536066, 13.30307531618842, 13.60586568351050