| L(s) = 1 | + 3-s + 3·5-s − 2·7-s + 9-s − 5·13-s + 3·15-s + 3·17-s + 4·19-s − 2·21-s + 6·23-s + 4·25-s + 27-s + 9·29-s + 8·31-s − 6·35-s − 7·37-s − 5·39-s + 3·41-s + 10·43-s + 3·45-s + 12·47-s − 3·49-s + 3·51-s − 9·53-s + 4·57-s − 6·59-s − 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 0.755·7-s + 1/3·9-s − 1.38·13-s + 0.774·15-s + 0.727·17-s + 0.917·19-s − 0.436·21-s + 1.25·23-s + 4/5·25-s + 0.192·27-s + 1.67·29-s + 1.43·31-s − 1.01·35-s − 1.15·37-s − 0.800·39-s + 0.468·41-s + 1.52·43-s + 0.447·45-s + 1.75·47-s − 3/7·49-s + 0.420·51-s − 1.23·53-s + 0.529·57-s − 0.781·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.500740840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.500740840\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| 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
−9.516749899854155725703471299078, −9.041773708650795693787324011801, −7.86790100712246650630397908735, −7.07095104250481703055869492985, −6.28889129406541946054483038974, −5.38704042867126013389488507017, −4.57037087943899774744148065639, −3.02015897534321884242390574572, −2.62513008159023679503924759669, −1.19804685965959534182745953435,
1.19804685965959534182745953435, 2.62513008159023679503924759669, 3.02015897534321884242390574572, 4.57037087943899774744148065639, 5.38704042867126013389488507017, 6.28889129406541946054483038974, 7.07095104250481703055869492985, 7.86790100712246650630397908735, 9.041773708650795693787324011801, 9.516749899854155725703471299078
