| L(s) = 1 | − 1.81·3-s + 2.70·5-s − 2.58·7-s + 0.298·9-s − 0.772·11-s − 13-s − 4.90·15-s − 0.701·17-s − 4.40·19-s + 4.70·21-s − 3.63·23-s + 2.29·25-s + 4.90·27-s − 2·29-s + 5.95·31-s + 1.40·33-s − 6.99·35-s + 6.70·37-s + 1.81·39-s + 11.4·41-s − 10.0·43-s + 0.806·45-s + 9.31·47-s − 0.298·49-s + 1.27·51-s − 1.40·53-s − 2.08·55-s + ⋯ |
| L(s) = 1 | − 1.04·3-s + 1.20·5-s − 0.978·7-s + 0.0994·9-s − 0.232·11-s − 0.277·13-s − 1.26·15-s − 0.170·17-s − 1.01·19-s + 1.02·21-s − 0.757·23-s + 0.459·25-s + 0.944·27-s − 0.371·29-s + 1.06·31-s + 0.244·33-s − 1.18·35-s + 1.10·37-s + 0.290·39-s + 1.78·41-s − 1.53·43-s + 0.120·45-s + 1.35·47-s − 0.0426·49-s + 0.178·51-s − 0.192·53-s − 0.281·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.012692764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012692764\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.81T + 3T^{2} \) |
| 5 | \( 1 - 2.70T + 5T^{2} \) |
| 7 | \( 1 + 2.58T + 7T^{2} \) |
| 11 | \( 1 + 0.772T + 11T^{2} \) |
| 17 | \( 1 + 0.701T + 17T^{2} \) |
| 19 | \( 1 + 4.40T + 19T^{2} \) |
| 23 | \( 1 + 3.63T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 37 | \( 1 - 6.70T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 9.31T + 47T^{2} \) |
| 53 | \( 1 + 1.40T + 53T^{2} \) |
| 59 | \( 1 + 2.85T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 1.04T + 71T^{2} \) |
| 73 | \( 1 - 7.40T + 73T^{2} \) |
| 79 | \( 1 + 2.08T + 79T^{2} \) |
| 83 | \( 1 + 9.58T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.765741912994971635061156263298, −7.84234262896305837584490307222, −6.73656940142883370742519549078, −6.21273137506061259961896945250, −5.85006222266756882753046868231, −5.00856477736663361265122724677, −4.11229436780406513063172081679, −2.84197914940038156117632940685, −2.07668217987781491844126476063, −0.60732108621601369533698405947,
0.60732108621601369533698405947, 2.07668217987781491844126476063, 2.84197914940038156117632940685, 4.11229436780406513063172081679, 5.00856477736663361265122724677, 5.85006222266756882753046868231, 6.21273137506061259961896945250, 6.73656940142883370742519549078, 7.84234262896305837584490307222, 8.765741912994971635061156263298
