| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 5·5-s − 6·6-s + 10·7-s + 8·8-s + 9·9-s − 10·10-s + 44·11-s − 12·12-s + 59·13-s + 20·14-s + 15·15-s + 16·16-s − 46·17-s + 18·18-s − 34·19-s − 20·20-s − 30·21-s + 88·22-s + 6·23-s − 24·24-s + 25·25-s + 118·26-s − 27·27-s + 40·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.539·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 1.25·13-s + 0.381·14-s + 0.258·15-s + 1/4·16-s − 0.656·17-s + 0.235·18-s − 0.410·19-s − 0.223·20-s − 0.311·21-s + 0.852·22-s + 0.0543·23-s − 0.204·24-s + 1/5·25-s + 0.890·26-s − 0.192·27-s + 0.269·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.104826695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.104826695\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 37 | \( 1 - p T \) |
| good | 7 | \( 1 - 10 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 59 T + p^{3} T^{2} \) |
| 17 | \( 1 + 46 T + p^{3} T^{2} \) |
| 19 | \( 1 + 34 T + p^{3} T^{2} \) |
| 23 | \( 1 - 6 T + p^{3} T^{2} \) |
| 29 | \( 1 - 7 T + p^{3} T^{2} \) |
| 31 | \( 1 + 182 T + p^{3} T^{2} \) |
| 41 | \( 1 - 360 T + p^{3} T^{2} \) |
| 43 | \( 1 - 101 T + p^{3} T^{2} \) |
| 47 | \( 1 + 35 T + p^{3} T^{2} \) |
| 53 | \( 1 + 507 T + p^{3} T^{2} \) |
| 59 | \( 1 - 821 T + p^{3} T^{2} \) |
| 61 | \( 1 - 70 T + p^{3} T^{2} \) |
| 67 | \( 1 - 612 T + p^{3} T^{2} \) |
| 71 | \( 1 - 88 T + p^{3} T^{2} \) |
| 73 | \( 1 - 622 T + p^{3} T^{2} \) |
| 79 | \( 1 - 8 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1223 T + p^{3} T^{2} \) |
| 89 | \( 1 - 345 T + p^{3} T^{2} \) |
| 97 | \( 1 - 870 T + p^{3} T^{2} \) |
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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
−9.423024744116603986507452653974, −8.604932737340222053356867888516, −7.67744513347574988284630534461, −6.65620952767566269563834297947, −6.15374719408614556561358194587, −5.09652429148215084293647153013, −4.17569626603987936486492610908, −3.59545060263045443917002701630, −1.98852534894635374816751593765, −0.894283755977934126501849682698,
0.894283755977934126501849682698, 1.98852534894635374816751593765, 3.59545060263045443917002701630, 4.17569626603987936486492610908, 5.09652429148215084293647153013, 6.15374719408614556561358194587, 6.65620952767566269563834297947, 7.67744513347574988284630534461, 8.604932737340222053356867888516, 9.423024744116603986507452653974
