| L(s) = 1 | − 37·7-s + 89·13-s − 163·19-s − 289·31-s + 110·37-s + 71·43-s + 1.02e3·49-s + 719·61-s + 1.00e3·67-s + 1.19e3·73-s + 884·79-s − 3.29e3·91-s − 523·97-s + 1.82e3·103-s − 1.56e3·109-s + ⋯ |
| L(s) = 1 | − 1.99·7-s + 1.89·13-s − 1.96·19-s − 1.67·31-s + 0.488·37-s + 0.251·43-s + 2.99·49-s + 1.50·61-s + 1.83·67-s + 1.90·73-s + 1.25·79-s − 3.79·91-s − 0.547·97-s + 1.74·103-s − 1.37·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.269881474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.269881474\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 37 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 89 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + 163 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + 289 T + p^{3} T^{2} \) |
| 37 | \( 1 - 110 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 71 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 719 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1007 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 1190 T + p^{3} T^{2} \) |
| 79 | \( 1 - 884 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + 523 T + p^{3} T^{2} \) |
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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
−9.636386008267152287572161488863, −8.967235521470196287145130780374, −8.218682802472888708045418488240, −6.83146636617729953575209298349, −6.38106031670799899939754245305, −5.63392505457052368141727963109, −3.96668476671684627007655389129, −3.51708882647338372716903051736, −2.22284650787630237148287589882, −0.59375312325564301977108165987,
0.59375312325564301977108165987, 2.22284650787630237148287589882, 3.51708882647338372716903051736, 3.96668476671684627007655389129, 5.63392505457052368141727963109, 6.38106031670799899939754245305, 6.83146636617729953575209298349, 8.218682802472888708045418488240, 8.967235521470196287145130780374, 9.636386008267152287572161488863
