| L(s) = 1 | − 0.193·2-s + 3-s − 1.96·4-s − 0.193·6-s + 7-s + 0.768·8-s + 9-s + 2·11-s − 1.96·12-s − 1.35·13-s − 0.193·14-s + 3.77·16-s + 3.35·17-s − 0.193·18-s + 5.35·19-s + 21-s − 0.387·22-s − 4.96·23-s + 0.768·24-s + 0.261·26-s + 27-s − 1.96·28-s + 7.92·29-s + 4.57·31-s − 2.26·32-s + 2·33-s − 0.649·34-s + ⋯ |
| L(s) = 1 | − 0.137·2-s + 0.577·3-s − 0.981·4-s − 0.0791·6-s + 0.377·7-s + 0.271·8-s + 0.333·9-s + 0.603·11-s − 0.566·12-s − 0.374·13-s − 0.0518·14-s + 0.943·16-s + 0.812·17-s − 0.0457·18-s + 1.22·19-s + 0.218·21-s − 0.0826·22-s − 1.03·23-s + 0.156·24-s + 0.0513·26-s + 0.192·27-s − 0.370·28-s + 1.47·29-s + 0.821·31-s − 0.401·32-s + 0.348·33-s − 0.111·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.454583486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.454583486\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 0.193T + 2T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 - 3.35T + 17T^{2} \) |
| 19 | \( 1 - 5.35T + 19T^{2} \) |
| 23 | \( 1 + 4.96T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 - 4.57T + 31T^{2} \) |
| 37 | \( 1 - 0.775T + 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 + 9.92T + 47T^{2} \) |
| 53 | \( 1 + 8.57T + 53T^{2} \) |
| 59 | \( 1 + 8.62T + 59T^{2} \) |
| 61 | \( 1 + 8.70T + 61T^{2} \) |
| 67 | \( 1 + 9.92T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 9.35T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 3.22T + 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−10.61797598792780120314623660474, −9.697774530240902187425587204845, −9.240701171692835591998690599415, −8.095944194541780680855303610191, −7.68234739127162575410322514300, −6.22924475067329073486711598694, −5.02522262844140395814578506502, −4.15816000723335585044553795505, −3.00963324380550823526572834362, −1.22392924948101114786523316269,
1.22392924948101114786523316269, 3.00963324380550823526572834362, 4.15816000723335585044553795505, 5.02522262844140395814578506502, 6.22924475067329073486711598694, 7.68234739127162575410322514300, 8.095944194541780680855303610191, 9.240701171692835591998690599415, 9.697774530240902187425587204845, 10.61797598792780120314623660474
