L(s) = 1 | − 1.41·3-s − 2.82·5-s − 0.999·9-s + 6·11-s + 5.65·13-s + 4.00·15-s − 1.41·17-s + 4.24·19-s + 4·23-s + 3.00·25-s + 5.65·27-s − 6·29-s − 2.82·31-s − 8.48·33-s + 2·37-s − 8.00·39-s + 1.41·41-s + 10·43-s + 2.82·45-s + 2.82·47-s + 2.00·51-s − 2·53-s − 16.9·55-s − 6·57-s − 1.41·59-s + 8.48·61-s − 16.0·65-s + ⋯ |
L(s) = 1 | − 0.816·3-s − 1.26·5-s − 0.333·9-s + 1.80·11-s + 1.56·13-s + 1.03·15-s − 0.342·17-s + 0.973·19-s + 0.834·23-s + 0.600·25-s + 1.08·27-s − 1.11·29-s − 0.508·31-s − 1.47·33-s + 0.328·37-s − 1.28·39-s + 0.220·41-s + 1.52·43-s + 0.421·45-s + 0.412·47-s + 0.280·51-s − 0.274·53-s − 2.28·55-s − 0.794·57-s − 0.184·59-s + 1.08·61-s − 1.98·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9005590877\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9005590877\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 9.89T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.34111881659272468409509165154, −10.93555521595051586624466383266, −9.283022921718162149391426587714, −8.643598801295561321387078564966, −7.46277192085796072460255893369, −6.51654555533864875288148596534, −5.62438459772634997936091749812, −4.20627921744943583040989376989, −3.47217559487253245668620155024, −1.00981060999436537944740073766,
1.00981060999436537944740073766, 3.47217559487253245668620155024, 4.20627921744943583040989376989, 5.62438459772634997936091749812, 6.51654555533864875288148596534, 7.46277192085796072460255893369, 8.643598801295561321387078564966, 9.283022921718162149391426587714, 10.93555521595051586624466383266, 11.34111881659272468409509165154