| L(s) = 1 | + 2.93·3-s − 5.62i·7-s − 0.402·9-s − 17.4·11-s + 7.35·13-s + 25.3i·17-s + (2.59 − 18.8i)19-s − 16.4i·21-s + 19.3i·23-s − 27.5·27-s − 4.89i·29-s + 45.2i·31-s − 51.1·33-s + 25.2·37-s + 21.5·39-s + ⋯ |
| L(s) = 1 | + 0.977·3-s − 0.803i·7-s − 0.0446·9-s − 1.58·11-s + 0.565·13-s + 1.48i·17-s + (0.136 − 0.990i)19-s − 0.785i·21-s + 0.839i·23-s − 1.02·27-s − 0.168i·29-s + 1.45i·31-s − 1.55·33-s + 0.683·37-s + 0.553·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.278836787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.278836787\) |
| \(L(2)\) |
|
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 | \( 1 + (-2.59 + 18.8i)T \) |
| good | 3 | \( 1 - 2.93T + 9T^{2} \) |
| 7 | \( 1 + 5.62iT - 49T^{2} \) |
| 11 | \( 1 + 17.4T + 121T^{2} \) |
| 13 | \( 1 - 7.35T + 169T^{2} \) |
| 17 | \( 1 - 25.3iT - 289T^{2} \) |
| 23 | \( 1 - 19.3iT - 529T^{2} \) |
| 29 | \( 1 + 4.89iT - 841T^{2} \) |
| 31 | \( 1 - 45.2iT - 961T^{2} \) |
| 37 | \( 1 - 25.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 16.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 28.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 38.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 81.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 61.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 94.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 109.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 57.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 36.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 112. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 42.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 148.T + 9.40e3T^{2} \) |
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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
−9.173088568878832665706766888720, −8.316370204991929886192460597908, −7.913370863712684166423601042491, −7.14268824836210661137514500727, −6.08767012460326129265929499143, −5.19770709401723604469207682969, −4.16189373697734464017657063611, −3.29147558361390737864280232782, −2.54977148406848016098884824109, −1.31559356282308913235083154200,
0.27057686774930473395283663002, 2.10038294303513613253561360642, 2.71274204839040156331559537034, 3.48947258363975914374200857376, 4.76948679186302673027904670978, 5.54802233383292757926249677170, 6.32611049802712912700274170819, 7.70352139832783052181979211634, 7.908860288019820434735107561130, 8.793445155133057560966076288364
