| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s + 4·5-s − 4·6-s + 7-s + 9-s + 8·10-s + 3·11-s − 4·12-s + 2·13-s + 2·14-s − 8·15-s − 4·16-s − 2·17-s + 2·18-s + 19-s + 8·20-s − 2·21-s + 6·22-s + 11·25-s + 4·26-s + 4·27-s + 2·28-s + 8·29-s − 16·30-s − 5·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s + 1.78·5-s − 1.63·6-s + 0.377·7-s + 1/3·9-s + 2.52·10-s + 0.904·11-s − 1.15·12-s + 0.554·13-s + 0.534·14-s − 2.06·15-s − 16-s − 0.485·17-s + 0.471·18-s + 0.229·19-s + 1.78·20-s − 0.436·21-s + 1.27·22-s + 11/5·25-s + 0.784·26-s + 0.769·27-s + 0.377·28-s + 1.48·29-s − 2.92·30-s − 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.224266485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.224266485\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4013 | \( 1+O(T) \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.733307016292182201470530080278, −7.13195109010699529855883116032, −6.44847889143151588305592272408, −6.00987769206640373341160680249, −5.55374327593100525288801835894, −4.86892351817549506409155287668, −4.19801965133586384564074386093, −3.02865667935775685711300181536, −2.10489078586203324162174094622, −1.08014622018469319166471885891,
1.08014622018469319166471885891, 2.10489078586203324162174094622, 3.02865667935775685711300181536, 4.19801965133586384564074386093, 4.86892351817549506409155287668, 5.55374327593100525288801835894, 6.00987769206640373341160680249, 6.44847889143151588305592272408, 7.13195109010699529855883116032, 8.733307016292182201470530080278
