| L(s) = 1 | + 6·3-s + 8·5-s − 7·7-s + 9·9-s + 56·11-s − 28·13-s + 48·15-s − 90·17-s + 74·19-s − 42·21-s − 96·23-s − 61·25-s − 108·27-s − 222·29-s − 100·31-s + 336·33-s − 56·35-s + 58·37-s − 168·39-s + 422·41-s + 512·43-s + 72·45-s + 148·47-s + 49·49-s − 540·51-s − 642·53-s + 448·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.715·5-s − 0.377·7-s + 1/3·9-s + 1.53·11-s − 0.597·13-s + 0.826·15-s − 1.28·17-s + 0.893·19-s − 0.436·21-s − 0.870·23-s − 0.487·25-s − 0.769·27-s − 1.42·29-s − 0.579·31-s + 1.77·33-s − 0.270·35-s + 0.257·37-s − 0.689·39-s + 1.60·41-s + 1.81·43-s + 0.238·45-s + 0.459·47-s + 1/7·49-s − 1.48·51-s − 1.66·53-s + 1.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.013227789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.013227789\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 5 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 56 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 90 T + p^{3} T^{2} \) |
| 19 | \( 1 - 74 T + p^{3} T^{2} \) |
| 23 | \( 1 + 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 222 T + p^{3} T^{2} \) |
| 31 | \( 1 + 100 T + p^{3} T^{2} \) |
| 37 | \( 1 - 58 T + p^{3} T^{2} \) |
| 41 | \( 1 - 422 T + p^{3} T^{2} \) |
| 43 | \( 1 - 512 T + p^{3} T^{2} \) |
| 47 | \( 1 - 148 T + p^{3} T^{2} \) |
| 53 | \( 1 + 642 T + p^{3} T^{2} \) |
| 59 | \( 1 + 318 T + p^{3} T^{2} \) |
| 61 | \( 1 - 720 T + p^{3} T^{2} \) |
| 67 | \( 1 + 412 T + p^{3} T^{2} \) |
| 71 | \( 1 - 448 T + p^{3} T^{2} \) |
| 73 | \( 1 - 994 T + p^{3} T^{2} \) |
| 79 | \( 1 + 296 T + p^{3} T^{2} \) |
| 83 | \( 1 - 386 T + p^{3} T^{2} \) |
| 89 | \( 1 + 6 T + p^{3} T^{2} \) |
| 97 | \( 1 + 138 T + p^{3} T^{2} \) |
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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
−14.42490932397970822854253744862, −13.94540155099789204265027137159, −12.76730423850578329433656751386, −11.32922960889032191166477063688, −9.492464754369934725311024839254, −9.185429752254765877102764611050, −7.53102564612712724747366180675, −6.05529226935475292475886002197, −3.89173908277123523252019440345, −2.17798397659288933598236518951,
2.17798397659288933598236518951, 3.89173908277123523252019440345, 6.05529226935475292475886002197, 7.53102564612712724747366180675, 9.185429752254765877102764611050, 9.492464754369934725311024839254, 11.32922960889032191166477063688, 12.76730423850578329433656751386, 13.94540155099789204265027137159, 14.42490932397970822854253744862
