| L(s) = 1 | − 2·3-s + 9·5-s + 31·7-s − 23·9-s + 57·11-s + 52·13-s − 18·15-s + 69·17-s + 19·19-s − 62·21-s + 72·23-s − 44·25-s + 100·27-s + 150·29-s − 32·31-s − 114·33-s + 279·35-s + 226·37-s − 104·39-s − 258·41-s − 67·43-s − 207·45-s − 579·47-s + 618·49-s − 138·51-s + 432·53-s + 513·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 0.804·5-s + 1.67·7-s − 0.851·9-s + 1.56·11-s + 1.10·13-s − 0.309·15-s + 0.984·17-s + 0.229·19-s − 0.644·21-s + 0.652·23-s − 0.351·25-s + 0.712·27-s + 0.960·29-s − 0.185·31-s − 0.601·33-s + 1.34·35-s + 1.00·37-s − 0.427·39-s − 0.982·41-s − 0.237·43-s − 0.685·45-s − 1.79·47-s + 1.80·49-s − 0.378·51-s + 1.11·53-s + 1.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.303423688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.303423688\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 7 | \( 1 - 31 T + p^{3} T^{2} \) |
| 11 | \( 1 - 57 T + p^{3} T^{2} \) |
| 13 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 69 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 150 T + p^{3} T^{2} \) |
| 31 | \( 1 + 32 T + p^{3} T^{2} \) |
| 37 | \( 1 - 226 T + p^{3} T^{2} \) |
| 41 | \( 1 + 258 T + p^{3} T^{2} \) |
| 43 | \( 1 + 67 T + p^{3} T^{2} \) |
| 47 | \( 1 + 579 T + p^{3} T^{2} \) |
| 53 | \( 1 - 432 T + p^{3} T^{2} \) |
| 59 | \( 1 + 330 T + p^{3} T^{2} \) |
| 61 | \( 1 - 13 T + p^{3} T^{2} \) |
| 67 | \( 1 + 856 T + p^{3} T^{2} \) |
| 71 | \( 1 + 642 T + p^{3} T^{2} \) |
| 73 | \( 1 + 487 T + p^{3} T^{2} \) |
| 79 | \( 1 - 700 T + p^{3} T^{2} \) |
| 83 | \( 1 + 12 T + p^{3} T^{2} \) |
| 89 | \( 1 + 600 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1424 T + p^{3} T^{2} \) |
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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
−9.229985589685166536975813554519, −8.576711183071378598327540646622, −7.892931274873082622035056184589, −6.66201777212695273355694321252, −5.93080292704206198272499623702, −5.24784188512278338190510622228, −4.30688584410465224727702433152, −3.12733224881032251138485402995, −1.65006164214392016088048213028, −1.10702451822723800626756830215,
1.10702451822723800626756830215, 1.65006164214392016088048213028, 3.12733224881032251138485402995, 4.30688584410465224727702433152, 5.24784188512278338190510622228, 5.93080292704206198272499623702, 6.66201777212695273355694321252, 7.892931274873082622035056184589, 8.576711183071378598327540646622, 9.229985589685166536975813554519
