| L(s) = 1 | − 3·3-s − 5·5-s + 9·9-s + 15·15-s − 14·17-s − 22·19-s − 34·23-s + 25·25-s − 27·27-s − 2·31-s − 45·45-s + 14·47-s + 49·49-s + 42·51-s + 86·53-s + 66·57-s + 118·61-s + 102·69-s − 75·75-s − 98·79-s + 81·81-s + 154·83-s + 70·85-s + 6·93-s + 110·95-s + 106·107-s + 22·109-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s + 9-s + 15-s − 0.823·17-s − 1.15·19-s − 1.47·23-s + 25-s − 27-s − 0.0645·31-s − 45-s + 0.297·47-s + 49-s + 0.823·51-s + 1.62·53-s + 1.15·57-s + 1.93·61-s + 1.47·69-s − 75-s − 1.24·79-s + 81-s + 1.85·83-s + 0.823·85-s + 2/31·93-s + 1.15·95-s + 0.990·107-s + 0.201·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6919924345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6919924345\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 + 14 T + p^{2} T^{2} \) |
| 19 | \( 1 + 22 T + p^{2} T^{2} \) |
| 23 | \( 1 + 34 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 2 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 - 14 T + p^{2} T^{2} \) |
| 53 | \( 1 - 86 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 118 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 + 98 T + p^{2} T^{2} \) |
| 83 | \( 1 - 154 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−10.12066638776399898870028761045, −8.913307505590447755838218865362, −8.104525028541673877709977755651, −7.16880196204179971371658981063, −6.46431643464178499659604753073, −5.51048942175738919327648925266, −4.39553042758654672309550823054, −3.88362436103542259734881832759, −2.17406804061546681166082789688, −0.53196147649580727530295040820,
0.53196147649580727530295040820, 2.17406804061546681166082789688, 3.88362436103542259734881832759, 4.39553042758654672309550823054, 5.51048942175738919327648925266, 6.46431643464178499659604753073, 7.16880196204179971371658981063, 8.104525028541673877709977755651, 8.913307505590447755838218865362, 10.12066638776399898870028761045
