| L(s) = 1 | − 0.372i·3-s − 3.51i·7-s + 26.8·9-s − 21.1·11-s − 14.0i·13-s − 17.2i·17-s + 19·19-s − 1.30·21-s + 171. i·23-s − 20.0i·27-s − 264.·29-s + 185.·31-s + 7.86i·33-s + 212. i·37-s − 5.24·39-s + ⋯ |
| L(s) = 1 | − 0.0716i·3-s − 0.189i·7-s + 0.994·9-s − 0.579·11-s − 0.300i·13-s − 0.245i·17-s + 0.229·19-s − 0.0135·21-s + 1.55i·23-s − 0.142i·27-s − 1.69·29-s + 1.07·31-s + 0.0415i·33-s + 0.946i·37-s − 0.0215·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.783378160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.783378160\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 + 0.372iT - 27T^{2} \) |
| 7 | \( 1 + 3.51iT - 343T^{2} \) |
| 11 | \( 1 + 21.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 17.2iT - 4.91e3T^{2} \) |
| 23 | \( 1 - 171. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 264.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 212. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 157.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 258. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 293. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 215. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 537.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 280.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 147. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 913.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 678. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 608.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 282. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 214.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.67e3iT - 9.12e5T^{2} \) |
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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
−9.069191445235326050861703838039, −8.030827361211265385006724645094, −7.46017250445257601583530278343, −6.78463835857084639537009030327, −5.70227832510578812644076185508, −4.99679236609114953644374533669, −4.01408167737283876288449496484, −3.15759520499613650040878052453, −1.95027329829287005412447894770, −0.956144762180761604091990959191,
0.42312241808437623050202799728, 1.71690714576729488530511716762, 2.64459952375939956105287643850, 3.85225537827942747335644552577, 4.56279733457070445615248073564, 5.47735272200369880913907269576, 6.37241498582208450220784243622, 7.20458799190113058250041610841, 7.85861960088257865537163016462, 8.797790329239218308479451801498
