| L(s) = 1 | − 3-s − 2·4-s + 3.46·5-s + 1.73·7-s + 9-s − 3.46·11-s + 2·12-s − 3.46·15-s + 4·16-s + 3.46·19-s − 6.92·20-s − 1.73·21-s + 6·23-s + 6.99·25-s − 27-s − 3.46·28-s + 6·29-s + 1.73·31-s + 3.46·33-s + 5.99·35-s − 2·36-s + 6.92·41-s + 43-s + 6.92·44-s + 3.46·45-s − 3.46·47-s − 4·48-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1.54·5-s + 0.654·7-s + 0.333·9-s − 1.04·11-s + 0.577·12-s − 0.894·15-s + 16-s + 0.794·19-s − 1.54·20-s − 0.377·21-s + 1.25·23-s + 1.39·25-s − 0.192·27-s − 0.654·28-s + 1.11·29-s + 0.311·31-s + 0.603·33-s + 1.01·35-s − 0.333·36-s + 1.08·41-s + 0.152·43-s + 1.04·44-s + 0.516·45-s − 0.505·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.309732006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.309732006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 8.66T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 1.73T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 + 5.19T + 97T^{2} \) |
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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
−10.61830362665621724149108319238, −10.09112703417803556791200386879, −9.267897418273464623679307917199, −8.391172421529893030852853557508, −7.24963530321341423501378290668, −5.93064261575568266342607788636, −5.28045701610837857299657547026, −4.61275172251397988724758301190, −2.78409310180320161668009725310, −1.19333478724103673445354913370,
1.19333478724103673445354913370, 2.78409310180320161668009725310, 4.61275172251397988724758301190, 5.28045701610837857299657547026, 5.93064261575568266342607788636, 7.24963530321341423501378290668, 8.391172421529893030852853557508, 9.267897418273464623679307917199, 10.09112703417803556791200386879, 10.61830362665621724149108319238
