| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 6·6-s + 7·7-s + 8·8-s + 9·9-s + 43.3·11-s + 12·12-s + 48.3·13-s + 14·14-s + 16·16-s − 10.3·17-s + 18·18-s + 20·19-s + 21·21-s + 86.6·22-s − 35.6·23-s + 24·24-s + 96.6·26-s + 27·27-s + 28·28-s + 15·29-s − 139.·31-s + 32·32-s + 129.·33-s − 20.6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.18·11-s + 0.288·12-s + 1.03·13-s + 0.267·14-s + 0.250·16-s − 0.146·17-s + 0.235·18-s + 0.241·19-s + 0.218·21-s + 0.839·22-s − 0.322·23-s + 0.204·24-s + 0.728·26-s + 0.192·27-s + 0.188·28-s + 0.0960·29-s − 0.808·31-s + 0.176·32-s + 0.685·33-s − 0.103·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.150364623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.150364623\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 11 | \( 1 - 43.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20T + 6.85e3T^{2} \) |
| 23 | \( 1 + 35.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 15T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 160.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 4.30T + 6.89e4T^{2} \) |
| 43 | \( 1 - 11.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 113.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 299.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 483.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 130.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 10.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 547.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 792.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 348.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.62e3T + 9.12e5T^{2} \) |
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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
−9.384209149153802878699045289411, −8.724279544482992870423691968008, −7.85023453891225184354690763025, −6.90084770454488505220819449241, −6.15480640798197358314340062332, −5.12944687173069531530088609792, −4.01655751090288225410899694881, −3.50281935797353101414118265659, −2.15291524982192090615010662059, −1.16116638280892497217441415263,
1.16116638280892497217441415263, 2.15291524982192090615010662059, 3.50281935797353101414118265659, 4.01655751090288225410899694881, 5.12944687173069531530088609792, 6.15480640798197358314340062332, 6.90084770454488505220819449241, 7.85023453891225184354690763025, 8.724279544482992870423691968008, 9.384209149153802878699045289411
