| L(s) = 1 | + 6.86·3-s − 5·5-s + 15.2·7-s + 20.1·9-s + 11·11-s + 30.1·13-s − 34.3·15-s + 74.4·17-s + 30.6·19-s + 104.·21-s + 111.·23-s + 25·25-s − 46.7·27-s + 23.5·29-s − 272.·31-s + 75.5·33-s − 76.2·35-s + 292.·37-s + 207.·39-s − 127.·41-s − 466.·43-s − 100.·45-s + 430.·47-s − 110.·49-s + 511.·51-s + 235.·53-s − 55·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s − 0.447·5-s + 0.823·7-s + 0.747·9-s + 0.301·11-s + 0.644·13-s − 0.591·15-s + 1.06·17-s + 0.369·19-s + 1.08·21-s + 1.00·23-s + 0.200·25-s − 0.333·27-s + 0.150·29-s − 1.57·31-s + 0.398·33-s − 0.368·35-s + 1.29·37-s + 0.851·39-s − 0.485·41-s − 1.65·43-s − 0.334·45-s + 1.33·47-s − 0.321·49-s + 1.40·51-s + 0.611·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.886808413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.886808413\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 - 6.86T + 27T^{2} \) |
| 7 | \( 1 - 15.2T + 343T^{2} \) |
| 13 | \( 1 - 30.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 30.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 23.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 272.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 292.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 127.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 466.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 430.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 235.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 167.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 363.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 611.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 315.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 372.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 300.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 73.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.38e3T + 9.12e5T^{2} \) |
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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
−11.78502663196403586002873982917, −10.93540067866230902230804422960, −9.638258512862470882533990235489, −8.709980140552451853639224644898, −8.014790905396583384842067771247, −7.14221518940152372109385517232, −5.44309773024090185549521277792, −4.01070019206486856816248095041, −3.01072749270940355524403428897, −1.43389367634362484347706137116,
1.43389367634362484347706137116, 3.01072749270940355524403428897, 4.01070019206486856816248095041, 5.44309773024090185549521277792, 7.14221518940152372109385517232, 8.014790905396583384842067771247, 8.709980140552451853639224644898, 9.638258512862470882533990235489, 10.93540067866230902230804422960, 11.78502663196403586002873982917
