| L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s + 7·7-s − 8·8-s − 10·10-s − 30·11-s − 34·13-s − 14·14-s + 16·16-s − 30·17-s + 128·19-s + 20·20-s + 60·22-s + 210·23-s + 25·25-s + 68·26-s + 28·28-s − 216·29-s + 128·31-s − 32·32-s + 60·34-s + 35·35-s + 2·37-s − 256·38-s − 40·40-s − 234·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.822·11-s − 0.725·13-s − 0.267·14-s + 1/4·16-s − 0.428·17-s + 1.54·19-s + 0.223·20-s + 0.581·22-s + 1.90·23-s + 1/5·25-s + 0.512·26-s + 0.188·28-s − 1.38·29-s + 0.741·31-s − 0.176·32-s + 0.302·34-s + 0.169·35-s + 0.00888·37-s − 1.09·38-s − 0.158·40-s − 0.891·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.496324736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496324736\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 128 T + p^{3} T^{2} \) |
| 23 | \( 1 - 210 T + p^{3} T^{2} \) |
| 29 | \( 1 + 216 T + p^{3} T^{2} \) |
| 31 | \( 1 - 128 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 T + p^{3} T^{2} \) |
| 41 | \( 1 + 234 T + p^{3} T^{2} \) |
| 43 | \( 1 - 236 T + p^{3} T^{2} \) |
| 47 | \( 1 + 132 T + p^{3} T^{2} \) |
| 53 | \( 1 - 168 T + p^{3} T^{2} \) |
| 59 | \( 1 + 12 T + p^{3} T^{2} \) |
| 61 | \( 1 - 758 T + p^{3} T^{2} \) |
| 67 | \( 1 - 164 T + p^{3} T^{2} \) |
| 71 | \( 1 - 306 T + p^{3} T^{2} \) |
| 73 | \( 1 - 866 T + p^{3} T^{2} \) |
| 79 | \( 1 + 304 T + p^{3} T^{2} \) |
| 83 | \( 1 + 720 T + p^{3} T^{2} \) |
| 89 | \( 1 + 186 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1370 T + p^{3} T^{2} \) |
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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
−10.03373922731553805026368713172, −9.412578330996854963133015864971, −8.525693542011127245839936530257, −7.52416824601442051684227239899, −6.94040657182254173447666084631, −5.56959649059124842271980092579, −4.90435017940667295959102338522, −3.18722725827681313496359430077, −2.15667994074621311416517497621, −0.805184597784559850149009036013,
0.805184597784559850149009036013, 2.15667994074621311416517497621, 3.18722725827681313496359430077, 4.90435017940667295959102338522, 5.56959649059124842271980092579, 6.94040657182254173447666084631, 7.52416824601442051684227239899, 8.525693542011127245839936530257, 9.412578330996854963133015864971, 10.03373922731553805026368713172
