| L(s) = 1 | − 2.27·3-s + 1.34·5-s + 4.27·7-s + 2.15·9-s − 2.09·11-s + 1.12·13-s − 3.05·15-s + 1.88·17-s + 5.13·19-s − 9.69·21-s + 7.99·23-s − 3.18·25-s + 1.92·27-s − 10.4·31-s + 4.74·33-s + 5.75·35-s + 5.27·37-s − 2.56·39-s + 8.84·41-s + 4.60·43-s + 2.90·45-s − 6.92·47-s + 11.2·49-s − 4.27·51-s + 2.66·53-s − 2.81·55-s − 11.6·57-s + ⋯ |
| L(s) = 1 | − 1.31·3-s + 0.602·5-s + 1.61·7-s + 0.718·9-s − 0.630·11-s + 0.313·13-s − 0.789·15-s + 0.457·17-s + 1.17·19-s − 2.11·21-s + 1.66·23-s − 0.637·25-s + 0.369·27-s − 1.88·31-s + 0.826·33-s + 0.971·35-s + 0.866·37-s − 0.410·39-s + 1.38·41-s + 0.702·43-s + 0.432·45-s − 1.00·47-s + 1.60·49-s − 0.599·51-s + 0.366·53-s − 0.379·55-s − 1.54·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.672939707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.672939707\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2.27T + 3T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 7 | \( 1 - 4.27T + 7T^{2} \) |
| 11 | \( 1 + 2.09T + 11T^{2} \) |
| 13 | \( 1 - 1.12T + 13T^{2} \) |
| 17 | \( 1 - 1.88T + 17T^{2} \) |
| 19 | \( 1 - 5.13T + 19T^{2} \) |
| 23 | \( 1 - 7.99T + 23T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 5.27T + 37T^{2} \) |
| 41 | \( 1 - 8.84T + 41T^{2} \) |
| 43 | \( 1 - 4.60T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 2.66T + 53T^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 + 4.74T + 61T^{2} \) |
| 67 | \( 1 - 2.12T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 5.77T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 5.47T + 89T^{2} \) |
| 97 | \( 1 - 3.31T + 97T^{2} \) |
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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
−8.633417023875448126474319720309, −7.60728624160091631795471692101, −7.28505395907785779717254047018, −6.04325692012904699085454306141, −5.48280478654910658004620028604, −5.12377293132416255837193612619, −4.28811605691114123910272248701, −2.94090281492610307155390447459, −1.72166367943700141674059340131, −0.885820278156686722762651985519,
0.885820278156686722762651985519, 1.72166367943700141674059340131, 2.94090281492610307155390447459, 4.28811605691114123910272248701, 5.12377293132416255837193612619, 5.48280478654910658004620028604, 6.04325692012904699085454306141, 7.28505395907785779717254047018, 7.60728624160091631795471692101, 8.633417023875448126474319720309
