| L(s) = 1 | + 5-s − 4·11-s − 6·13-s + 6·17-s − 4·19-s + 23-s + 25-s + 6·29-s + 8·31-s + 6·37-s − 10·41-s − 4·43-s − 8·47-s − 7·49-s + 14·53-s − 4·55-s + 10·61-s − 6·65-s − 4·67-s + 8·71-s + 2·73-s + 12·79-s − 16·83-s + 6·85-s + 2·89-s − 4·95-s − 14·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s − 1.66·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s − 49-s + 1.92·53-s − 0.539·55-s + 1.28·61-s − 0.744·65-s − 0.488·67-s + 0.949·71-s + 0.234·73-s + 1.35·79-s − 1.75·83-s + 0.650·85-s + 0.211·89-s − 0.410·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.33053565572842, −15.54815137011599, −14.96848447722960, −14.68655999060296, −13.95691026778912, −13.44547807292438, −12.81995532661075, −12.35890615127446, −11.84744641216570, −11.17454914494992, −10.23429260474757, −10.00541138810414, −9.790887291458925, −8.615764171476395, −8.191791628567376, −7.653489945184931, −6.895196323970055, −6.377260626042192, −5.473937418123080, −5.025690928574172, −4.542467352943417, −3.413455845625358, −2.681460415685319, −2.243617246695483, −1.087434345016722, 0,
1.087434345016722, 2.243617246695483, 2.681460415685319, 3.413455845625358, 4.542467352943417, 5.025690928574172, 5.473937418123080, 6.377260626042192, 6.895196323970055, 7.653489945184931, 8.191791628567376, 8.615764171476395, 9.790887291458925, 10.00541138810414, 10.23429260474757, 11.17454914494992, 11.84744641216570, 12.35890615127446, 12.81995532661075, 13.44547807292438, 13.95691026778912, 14.68655999060296, 14.96848447722960, 15.54815137011599, 16.33053565572842
