L(s) = 1 | + 0.512·3-s − 5-s − 7-s − 2.73·9-s − 3.81·11-s + 13-s − 0.512·15-s − 0.761·17-s − 1.04·19-s − 0.512·21-s + 5.81·23-s + 25-s − 2.93·27-s + 4.32·29-s + 4.61·31-s − 1.95·33-s + 35-s − 5.92·37-s + 0.512·39-s − 7.43·41-s − 2.78·43-s + 2.73·45-s − 3.12·47-s + 49-s − 0.390·51-s + 7.12·53-s + 3.81·55-s + ⋯ |
L(s) = 1 | + 0.295·3-s − 0.447·5-s − 0.377·7-s − 0.912·9-s − 1.14·11-s + 0.277·13-s − 0.132·15-s − 0.184·17-s − 0.240·19-s − 0.111·21-s + 1.21·23-s + 0.200·25-s − 0.565·27-s + 0.802·29-s + 0.828·31-s − 0.339·33-s + 0.169·35-s − 0.973·37-s + 0.0819·39-s − 1.16·41-s − 0.425·43-s + 0.408·45-s − 0.455·47-s + 0.142·49-s − 0.0546·51-s + 0.978·53-s + 0.513·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.260936348\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.260936348\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 0.512T + 3T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 17 | \( 1 + 0.761T + 17T^{2} \) |
| 19 | \( 1 + 1.04T + 19T^{2} \) |
| 23 | \( 1 - 5.81T + 23T^{2} \) |
| 29 | \( 1 - 4.32T + 29T^{2} \) |
| 31 | \( 1 - 4.61T + 31T^{2} \) |
| 37 | \( 1 + 5.92T + 37T^{2} \) |
| 41 | \( 1 + 7.43T + 41T^{2} \) |
| 43 | \( 1 + 2.78T + 43T^{2} \) |
| 47 | \( 1 + 3.12T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 - 8.24T + 61T^{2} \) |
| 67 | \( 1 - 2.21T + 67T^{2} \) |
| 71 | \( 1 - 4.58T + 71T^{2} \) |
| 73 | \( 1 + 3.07T + 73T^{2} \) |
| 79 | \( 1 - 9.11T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−8.513395120477472370396689386882, −7.989711906123809179302647851853, −7.04575440582848409809558102924, −6.43194712137990054132741293418, −5.38697606373562795889220758544, −4.88788294310038060175131097773, −3.68489666453329605537091603861, −3.02112980269086870654108432361, −2.24864501631302080887951834213, −0.61835439471621721868960779322,
0.61835439471621721868960779322, 2.24864501631302080887951834213, 3.02112980269086870654108432361, 3.68489666453329605537091603861, 4.88788294310038060175131097773, 5.38697606373562795889220758544, 6.43194712137990054132741293418, 7.04575440582848409809558102924, 7.989711906123809179302647851853, 8.513395120477472370396689386882