| L(s) = 1 | − 3.37·3-s + 5-s + 7-s + 8.37·9-s + 0.627·11-s + 1.37·13-s − 3.37·15-s + 5.37·17-s + 6.74·19-s − 3.37·21-s − 6.74·23-s + 25-s − 18.1·27-s − 1.37·29-s + 8·31-s − 2.11·33-s + 35-s + 2·37-s − 4.62·39-s − 4.74·41-s + 2.74·43-s + 8.37·45-s − 10.1·47-s + 49-s − 18.1·51-s + 0.744·53-s + 0.627·55-s + ⋯ |
| L(s) = 1 | − 1.94·3-s + 0.447·5-s + 0.377·7-s + 2.79·9-s + 0.189·11-s + 0.380·13-s − 0.870·15-s + 1.30·17-s + 1.54·19-s − 0.735·21-s − 1.40·23-s + 0.200·25-s − 3.48·27-s − 0.254·29-s + 1.43·31-s − 0.368·33-s + 0.169·35-s + 0.328·37-s − 0.741·39-s − 0.740·41-s + 0.418·43-s + 1.24·45-s − 1.47·47-s + 0.142·49-s − 2.53·51-s + 0.102·53-s + 0.0846·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.171201381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171201381\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 3.37T + 3T^{2} \) |
| 11 | \( 1 - 0.627T + 11T^{2} \) |
| 13 | \( 1 - 1.37T + 13T^{2} \) |
| 17 | \( 1 - 5.37T + 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 23 | \( 1 + 6.74T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 4.74T + 41T^{2} \) |
| 43 | \( 1 - 2.74T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 0.744T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 8.74T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 2.11T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 3.25T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−9.356119409938334521500099127335, −7.989239271079669064278822545795, −7.38204745496586367244854668117, −6.37887338413301538411515404611, −5.89597733191680165733786868225, −5.22352100977765965601834883924, −4.53007556661439647468753530942, −3.43352607174891821077471741983, −1.67943254384597477611710126317, −0.843367691117737739692710094095,
0.843367691117737739692710094095, 1.67943254384597477611710126317, 3.43352607174891821077471741983, 4.53007556661439647468753530942, 5.22352100977765965601834883924, 5.89597733191680165733786868225, 6.37887338413301538411515404611, 7.38204745496586367244854668117, 7.989239271079669064278822545795, 9.356119409938334521500099127335
