| L(s) = 1 | + 0.867·2-s + 0.595·3-s − 1.24·4-s + 3.84·5-s + 0.516·6-s + 1.26·7-s − 2.81·8-s − 2.64·9-s + 3.33·10-s − 11-s − 0.742·12-s + 5.03·13-s + 1.09·14-s + 2.28·15-s + 0.0519·16-s − 3.67·17-s − 2.29·18-s + 3.68·19-s − 4.79·20-s + 0.755·21-s − 0.867·22-s + 5.90·23-s − 1.67·24-s + 9.77·25-s + 4.37·26-s − 3.36·27-s − 1.58·28-s + ⋯ |
| L(s) = 1 | + 0.613·2-s + 0.343·3-s − 0.623·4-s + 1.71·5-s + 0.210·6-s + 0.479·7-s − 0.995·8-s − 0.881·9-s + 1.05·10-s − 0.301·11-s − 0.214·12-s + 1.39·13-s + 0.293·14-s + 0.590·15-s + 0.0129·16-s − 0.890·17-s − 0.540·18-s + 0.845·19-s − 1.07·20-s + 0.164·21-s − 0.184·22-s + 1.23·23-s − 0.342·24-s + 1.95·25-s + 0.857·26-s − 0.646·27-s − 0.298·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.971037199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.971037199\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 - 0.867T + 2T^{2} \) |
| 3 | \( 1 - 0.595T + 3T^{2} \) |
| 5 | \( 1 - 3.84T + 5T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 13 | \( 1 - 5.03T + 13T^{2} \) |
| 17 | \( 1 + 3.67T + 17T^{2} \) |
| 19 | \( 1 - 3.68T + 19T^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 29 | \( 1 - 8.38T + 29T^{2} \) |
| 31 | \( 1 - 4.86T + 31T^{2} \) |
| 37 | \( 1 + 2.88T + 37T^{2} \) |
| 41 | \( 1 + 1.35T + 41T^{2} \) |
| 43 | \( 1 + 7.22T + 43T^{2} \) |
| 47 | \( 1 + 2.45T + 47T^{2} \) |
| 53 | \( 1 - 2.17T + 53T^{2} \) |
| 59 | \( 1 - 9.50T + 59T^{2} \) |
| 61 | \( 1 + 3.72T + 61T^{2} \) |
| 67 | \( 1 - 4.48T + 67T^{2} \) |
| 71 | \( 1 + 4.16T + 71T^{2} \) |
| 73 | \( 1 + 7.67T + 73T^{2} \) |
| 79 | \( 1 - 5.11T + 79T^{2} \) |
| 83 | \( 1 + 4.45T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 2.95T + 97T^{2} \) |
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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
−9.449852705612634286908671713648, −8.571222494561394418958432210622, −8.474940725971662482869395839250, −6.72777976163098320292799651421, −6.05593377016985594178631318728, −5.31519655149209342396795116088, −4.72105834835624900616251018583, −3.32072843355866764856951966710, −2.59411014527865336898488329153, −1.24631142996277733658440059584,
1.24631142996277733658440059584, 2.59411014527865336898488329153, 3.32072843355866764856951966710, 4.72105834835624900616251018583, 5.31519655149209342396795116088, 6.05593377016985594178631318728, 6.72777976163098320292799651421, 8.474940725971662482869395839250, 8.571222494561394418958432210622, 9.449852705612634286908671713648
