| L(s) = 1 | + 3-s + 9-s + 2·11-s + 2·13-s − 2·17-s − 4·19-s + 27-s − 2·29-s + 2·31-s + 2·33-s − 6·37-s + 2·39-s − 10·41-s − 8·43-s − 12·47-s − 7·49-s − 2·51-s + 8·53-s − 4·57-s − 10·59-s − 10·61-s + 8·67-s − 4·73-s + 10·79-s + 81-s − 4·83-s − 2·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.192·27-s − 0.371·29-s + 0.359·31-s + 0.348·33-s − 0.986·37-s + 0.320·39-s − 1.56·41-s − 1.21·43-s − 1.75·47-s − 49-s − 0.280·51-s + 1.09·53-s − 0.529·57-s − 1.30·59-s − 1.28·61-s + 0.977·67-s − 0.468·73-s + 1.12·79-s + 1/9·81-s − 0.439·83-s − 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.34606401607521008410348643942, −6.54690570892308640744621705983, −6.29547897566832954117196981452, −5.12728557639059850121908714994, −4.55100832862322076288565215506, −3.65516185355043188939899251971, −3.20248023259753786054913749878, −2.04931712235755675376871367260, −1.47384326310992183297540874451, 0,
1.47384326310992183297540874451, 2.04931712235755675376871367260, 3.20248023259753786054913749878, 3.65516185355043188939899251971, 4.55100832862322076288565215506, 5.12728557639059850121908714994, 6.29547897566832954117196981452, 6.54690570892308640744621705983, 7.34606401607521008410348643942
