| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.916722943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.916722943\) |
| \(L(\frac{5}{2})\) |
|
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
−10.81961654042097455382470414431, −9.332947477300739234223375928817, −8.926967538991871958888581492262, −8.075039826586910061568014625056, −7.19613463864932120751495021113, −6.13661158151388422844703701872, −4.48151947576344911418078296534, −3.74177329387695589513599401356, −2.60765318056494501320427177596, −1.11942682106930748214083529166,
1.11942682106930748214083529166, 2.60765318056494501320427177596, 3.74177329387695589513599401356, 4.48151947576344911418078296534, 6.13661158151388422844703701872, 7.19613463864932120751495021113, 8.075039826586910061568014625056, 8.926967538991871958888581492262, 9.332947477300739234223375928817, 10.81961654042097455382470414431
