| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 7-s − 3·8-s − 2·9-s + 4·11-s − 12-s − 13-s − 14-s − 16-s − 2·18-s − 6·19-s − 21-s + 4·22-s − 3·24-s − 26-s − 5·27-s + 28-s + 7·31-s + 5·32-s + 4·33-s + 2·36-s + 4·37-s − 6·38-s − 39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.377·7-s − 1.06·8-s − 2/3·9-s + 1.20·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s − 1/4·16-s − 0.471·18-s − 1.37·19-s − 0.218·21-s + 0.852·22-s − 0.612·24-s − 0.196·26-s − 0.962·27-s + 0.188·28-s + 1.25·31-s + 0.883·32-s + 0.696·33-s + 1/3·36-s + 0.657·37-s − 0.973·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.226693672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.226693672\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.137336175370570455158370286211, −7.11506758484051445834837725606, −6.18382323461402075157966073920, −6.03879953647401353841158787955, −4.88611047485524815987471444871, −4.28031132843975574213598865876, −3.62547166449923999392099280690, −2.92386518865550189156320772474, −2.10237647492739208778648010240, −0.64079892155178707280465812390,
0.64079892155178707280465812390, 2.10237647492739208778648010240, 2.92386518865550189156320772474, 3.62547166449923999392099280690, 4.28031132843975574213598865876, 4.88611047485524815987471444871, 6.03879953647401353841158787955, 6.18382323461402075157966073920, 7.11506758484051445834837725606, 8.137336175370570455158370286211
