| L(s) = 1 | + 2.89·3-s − 1.96·5-s + 2.82·7-s + 5.39·9-s − 5.91·11-s − 5.24·13-s − 5.68·15-s − 0.885·17-s − 6.74·19-s + 8.19·21-s + 5.73·23-s − 1.14·25-s + 6.93·27-s − 9.96·29-s + 0.549·31-s − 17.1·33-s − 5.55·35-s + 6.03·37-s − 15.1·39-s + 2.81·41-s + 11.4·43-s − 10.5·45-s − 8.25·47-s + 1.00·49-s − 2.56·51-s − 4.66·53-s + 11.6·55-s + ⋯ |
| L(s) = 1 | + 1.67·3-s − 0.878·5-s + 1.06·7-s + 1.79·9-s − 1.78·11-s − 1.45·13-s − 1.46·15-s − 0.214·17-s − 1.54·19-s + 1.78·21-s + 1.19·23-s − 0.228·25-s + 1.33·27-s − 1.84·29-s + 0.0986·31-s − 2.98·33-s − 0.938·35-s + 0.991·37-s − 2.43·39-s + 0.440·41-s + 1.74·43-s − 1.57·45-s − 1.20·47-s + 0.142·49-s − 0.359·51-s − 0.641·53-s + 1.56·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 2.89T + 3T^{2} \) |
| 5 | \( 1 + 1.96T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 5.91T + 11T^{2} \) |
| 13 | \( 1 + 5.24T + 13T^{2} \) |
| 17 | \( 1 + 0.885T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 + 9.96T + 29T^{2} \) |
| 31 | \( 1 - 0.549T + 31T^{2} \) |
| 37 | \( 1 - 6.03T + 37T^{2} \) |
| 41 | \( 1 - 2.81T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 8.25T + 47T^{2} \) |
| 53 | \( 1 + 4.66T + 53T^{2} \) |
| 59 | \( 1 + 1.78T + 59T^{2} \) |
| 61 | \( 1 + 7.68T + 61T^{2} \) |
| 67 | \( 1 + 2.96T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 + 9.57T + 83T^{2} \) |
| 89 | \( 1 - 7.66T + 89T^{2} \) |
| 97 | \( 1 + 2.26T + 97T^{2} \) |
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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
−7.979198925886640078698938747910, −7.57936222424151518243253623620, −7.24546155777033393144872651437, −5.71450810562417013467575537516, −4.59744906360915482849566828158, −4.42543645056434916126860688667, −3.20253727978108995879505055945, −2.50487744917393768426248951497, −1.88867956638371849858764491027, 0,
1.88867956638371849858764491027, 2.50487744917393768426248951497, 3.20253727978108995879505055945, 4.42543645056434916126860688667, 4.59744906360915482849566828158, 5.71450810562417013467575537516, 7.24546155777033393144872651437, 7.57936222424151518243253623620, 7.979198925886640078698938747910
