L(s) = 1 | + 3·5-s + 3·11-s + 2·13-s + 6·17-s − 2·19-s + 6·23-s + 4·25-s + 9·29-s + 7·31-s − 10·37-s + 4·43-s − 12·47-s − 3·53-s + 9·55-s + 3·59-s − 4·61-s + 6·65-s − 2·67-s + 2·73-s − 5·79-s − 9·83-s + 18·85-s − 6·89-s − 6·95-s − 13·97-s + 6·101-s + 16·103-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 0.904·11-s + 0.554·13-s + 1.45·17-s − 0.458·19-s + 1.25·23-s + 4/5·25-s + 1.67·29-s + 1.25·31-s − 1.64·37-s + 0.609·43-s − 1.75·47-s − 0.412·53-s + 1.21·55-s + 0.390·59-s − 0.512·61-s + 0.744·65-s − 0.244·67-s + 0.234·73-s − 0.562·79-s − 0.987·83-s + 1.95·85-s − 0.635·89-s − 0.615·95-s − 1.31·97-s + 0.597·101-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.404865909\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.404865909\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.106970234141479305794998979857, −6.98445836567590536119347277811, −6.51115643358981370975176094997, −5.90932841910865615916036178388, −5.19668251283389337272698980422, −4.48321902195496152445728641454, −3.39539046306628889226611627255, −2.77419035194339692578816188604, −1.61346728744823563189174744288, −1.06566635835283641561279708362,
1.06566635835283641561279708362, 1.61346728744823563189174744288, 2.77419035194339692578816188604, 3.39539046306628889226611627255, 4.48321902195496152445728641454, 5.19668251283389337272698980422, 5.90932841910865615916036178388, 6.51115643358981370975176094997, 6.98445836567590536119347277811, 8.106970234141479305794998979857