| L(s) = 1 | + 1.97·2-s − 3·3-s − 4.09·4-s − 2.49·5-s − 5.92·6-s − 20.7·7-s − 23.9·8-s + 9·9-s − 4.93·10-s + 59.5·11-s + 12.2·12-s − 62.8·13-s − 41.0·14-s + 7.49·15-s − 14.4·16-s + 38.4·17-s + 17.7·18-s − 114.·19-s + 10.2·20-s + 62.3·21-s + 117.·22-s − 23·23-s + 71.7·24-s − 118.·25-s − 124.·26-s − 27·27-s + 85.1·28-s + ⋯ |
| L(s) = 1 | + 0.698·2-s − 0.577·3-s − 0.511·4-s − 0.223·5-s − 0.403·6-s − 1.12·7-s − 1.05·8-s + 0.333·9-s − 0.156·10-s + 1.63·11-s + 0.295·12-s − 1.34·13-s − 0.784·14-s + 0.128·15-s − 0.226·16-s + 0.549·17-s + 0.232·18-s − 1.37·19-s + 0.114·20-s + 0.648·21-s + 1.13·22-s − 0.208·23-s + 0.609·24-s − 0.950·25-s − 0.937·26-s − 0.192·27-s + 0.574·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4704005405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4704005405\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 23 | \( 1 + 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 1.97T + 8T^{2} \) |
| 5 | \( 1 + 2.49T + 125T^{2} \) |
| 7 | \( 1 + 20.7T + 343T^{2} \) |
| 11 | \( 1 - 59.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 38.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 114.T + 6.85e3T^{2} \) |
| 31 | \( 1 + 195.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 22.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 514.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 18.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 633.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 512.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 334.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 331.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 287.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 91.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 619.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 520.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 790.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 157.T + 9.12e5T^{2} \) |
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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
−8.976045166308840294190587665483, −8.008952175618278975632513796153, −6.75657137625680715520006885471, −6.49000908589133279434272396487, −5.54201915563084594495662765018, −4.72060607226831089701373426014, −3.86424569737343035577958396852, −3.34561446374966016251869817437, −1.87780926319065956222292680652, −0.28241568016363743991406699993,
0.28241568016363743991406699993, 1.87780926319065956222292680652, 3.34561446374966016251869817437, 3.86424569737343035577958396852, 4.72060607226831089701373426014, 5.54201915563084594495662765018, 6.49000908589133279434272396487, 6.75657137625680715520006885471, 8.008952175618278975632513796153, 8.976045166308840294190587665483
