L(s) = 1 | + 3-s + 5-s − 1.64·7-s + 9-s + 0.354·11-s + 3.64·13-s + 15-s + 19-s − 1.64·21-s − 2·23-s + 25-s + 27-s + 9.64·29-s + 2·31-s + 0.354·33-s − 1.64·35-s − 6.93·37-s + 3.64·39-s + 1.64·41-s + 4.93·43-s + 45-s − 6·47-s − 4.29·49-s + 4·53-s + 0.354·55-s + 57-s − 3.29·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.622·7-s + 0.333·9-s + 0.106·11-s + 1.01·13-s + 0.258·15-s + 0.229·19-s − 0.359·21-s − 0.417·23-s + 0.200·25-s + 0.192·27-s + 1.79·29-s + 0.359·31-s + 0.0616·33-s − 0.278·35-s − 1.14·37-s + 0.583·39-s + 0.257·41-s + 0.752·43-s + 0.149·45-s − 0.875·47-s − 0.613·49-s + 0.549·53-s + 0.0477·55-s + 0.132·57-s − 0.428·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.687712125\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.687712125\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 - 0.354T + 11T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 9.64T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 6.93T + 37T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 43 | \( 1 - 4.93T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 3.29T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−8.442396752394157986719624710511, −7.71332936677065818663580779064, −6.68380188688673030128796869597, −6.35845969825037222464822410375, −5.43414412069168753127154597723, −4.51807291121211765475816958586, −3.61543669127245862917558507276, −2.97344029339240830851385293651, −1.99921459068528355971127076153, −0.917318745951632453200851973070,
0.917318745951632453200851973070, 1.99921459068528355971127076153, 2.97344029339240830851385293651, 3.61543669127245862917558507276, 4.51807291121211765475816958586, 5.43414412069168753127154597723, 6.35845969825037222464822410375, 6.68380188688673030128796869597, 7.71332936677065818663580779064, 8.442396752394157986719624710511