L(s) = 1 | + 0.710·3-s + 4.59·7-s − 2.49·9-s − 3.91·11-s − 0.572·13-s − 0.232·17-s + 5.55·19-s + 3.26·21-s + 4.93·23-s − 3.90·27-s + 4.13·29-s + 3.49·31-s − 2.78·33-s − 5.41·37-s − 0.406·39-s + 10.4·41-s − 1.38·43-s − 0.920·47-s + 14.0·49-s − 0.165·51-s + 1.23·53-s + 3.94·57-s − 4.50·59-s − 11.6·61-s − 11.4·63-s + 2.95·67-s + 3.50·69-s + ⋯ |
L(s) = 1 | + 0.410·3-s + 1.73·7-s − 0.831·9-s − 1.18·11-s − 0.158·13-s − 0.0564·17-s + 1.27·19-s + 0.711·21-s + 1.02·23-s − 0.751·27-s + 0.767·29-s + 0.627·31-s − 0.484·33-s − 0.890·37-s − 0.0651·39-s + 1.62·41-s − 0.211·43-s − 0.134·47-s + 2.01·49-s − 0.0231·51-s + 0.169·53-s + 0.522·57-s − 0.586·59-s − 1.49·61-s − 1.44·63-s + 0.360·67-s + 0.421·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.775123655\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.775123655\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 0.710T + 3T^{2} \) |
| 7 | \( 1 - 4.59T + 7T^{2} \) |
| 11 | \( 1 + 3.91T + 11T^{2} \) |
| 13 | \( 1 + 0.572T + 13T^{2} \) |
| 17 | \( 1 + 0.232T + 17T^{2} \) |
| 19 | \( 1 - 5.55T + 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 - 3.49T + 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 1.38T + 43T^{2} \) |
| 47 | \( 1 + 0.920T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 + 4.50T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 2.95T + 67T^{2} \) |
| 71 | \( 1 + 3.20T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 9.61T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 7.25T + 89T^{2} \) |
| 97 | \( 1 + 8.31T + 97T^{2} \) |
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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
−7.70771788408757860969617906667, −7.33992876858213333193747100270, −6.20034299265654490649503150764, −5.33222881336350041348589742916, −5.05792483021796117913322691554, −4.34186310814213649840383803376, −3.15482459882431132752101276775, −2.67483088668300646057002736339, −1.78582458748539910249917493807, −0.78470304251297588239721788651,
0.78470304251297588239721788651, 1.78582458748539910249917493807, 2.67483088668300646057002736339, 3.15482459882431132752101276775, 4.34186310814213649840383803376, 5.05792483021796117913322691554, 5.33222881336350041348589742916, 6.20034299265654490649503150764, 7.33992876858213333193747100270, 7.70771788408757860969617906667