L(s) = 1 | + 1.46·3-s + 2.98·5-s − 5.21·7-s − 0.844·9-s + 3.09·11-s + 7.06·13-s + 4.37·15-s + 0.543·17-s − 19-s − 7.65·21-s − 3.14·23-s + 3.88·25-s − 5.64·27-s + 3.77·29-s + 1.66·31-s + 4.54·33-s − 15.5·35-s − 5.67·37-s + 10.3·39-s + 4.93·41-s − 1.27·43-s − 2.51·45-s + 5.98·47-s + 20.1·49-s + 0.797·51-s + 12.8·53-s + 9.22·55-s + ⋯ |
L(s) = 1 | + 0.847·3-s + 1.33·5-s − 1.96·7-s − 0.281·9-s + 0.933·11-s + 1.95·13-s + 1.12·15-s + 0.131·17-s − 0.229·19-s − 1.66·21-s − 0.655·23-s + 0.777·25-s − 1.08·27-s + 0.701·29-s + 0.298·31-s + 0.791·33-s − 2.62·35-s − 0.933·37-s + 1.66·39-s + 0.771·41-s − 0.194·43-s − 0.375·45-s + 0.872·47-s + 2.87·49-s + 0.111·51-s + 1.76·53-s + 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.114860644\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.114860644\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 - 1.46T + 3T^{2} \) |
| 5 | \( 1 - 2.98T + 5T^{2} \) |
| 7 | \( 1 + 5.21T + 7T^{2} \) |
| 11 | \( 1 - 3.09T + 11T^{2} \) |
| 13 | \( 1 - 7.06T + 13T^{2} \) |
| 17 | \( 1 - 0.543T + 17T^{2} \) |
| 23 | \( 1 + 3.14T + 23T^{2} \) |
| 29 | \( 1 - 3.77T + 29T^{2} \) |
| 31 | \( 1 - 1.66T + 31T^{2} \) |
| 37 | \( 1 + 5.67T + 37T^{2} \) |
| 41 | \( 1 - 4.93T + 41T^{2} \) |
| 43 | \( 1 + 1.27T + 43T^{2} \) |
| 47 | \( 1 - 5.98T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 + 0.337T + 61T^{2} \) |
| 67 | \( 1 - 5.48T + 67T^{2} \) |
| 71 | \( 1 + 6.31T + 71T^{2} \) |
| 73 | \( 1 - 5.67T + 73T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 9.94T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.540065272442551756363653061724, −7.18695762302237336830502425683, −6.48487239376443920104838270597, −6.01029531017987020552867593679, −5.66040651624623580581516948266, −4.01213179995221671137405967992, −3.57667940628129130671945201600, −2.82106250814113098122840058826, −2.03099826686555345893658831610, −0.902064802870166863175698802205,
0.902064802870166863175698802205, 2.03099826686555345893658831610, 2.82106250814113098122840058826, 3.57667940628129130671945201600, 4.01213179995221671137405967992, 5.66040651624623580581516948266, 6.01029531017987020552867593679, 6.48487239376443920104838270597, 7.18695762302237336830502425683, 8.540065272442551756363653061724