| L(s) = 1 | + 1.93·3-s + 5-s + 3.43·7-s + 0.736·9-s − 5.42·11-s − 0.444·13-s + 1.93·15-s − 3.65·17-s − 1.22·19-s + 6.64·21-s − 6.88·23-s + 25-s − 4.37·27-s + 0.384·29-s + 3.01·31-s − 10.4·33-s + 3.43·35-s − 9.74·37-s − 0.858·39-s − 2.82·41-s − 6.95·43-s + 0.736·45-s + 5.29·47-s + 4.82·49-s − 7.05·51-s − 4.14·53-s − 5.42·55-s + ⋯ |
| L(s) = 1 | + 1.11·3-s + 0.447·5-s + 1.29·7-s + 0.245·9-s − 1.63·11-s − 0.123·13-s + 0.499·15-s − 0.885·17-s − 0.279·19-s + 1.45·21-s − 1.43·23-s + 0.200·25-s − 0.841·27-s + 0.0714·29-s + 0.541·31-s − 1.82·33-s + 0.581·35-s − 1.60·37-s − 0.137·39-s − 0.441·41-s − 1.05·43-s + 0.109·45-s + 0.771·47-s + 0.688·49-s − 0.988·51-s − 0.569·53-s − 0.730·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 1.93T + 3T^{2} \) |
| 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 + 5.42T + 11T^{2} \) |
| 13 | \( 1 + 0.444T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 1.22T + 19T^{2} \) |
| 23 | \( 1 + 6.88T + 23T^{2} \) |
| 29 | \( 1 - 0.384T + 29T^{2} \) |
| 31 | \( 1 - 3.01T + 31T^{2} \) |
| 37 | \( 1 + 9.74T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 + 6.95T + 43T^{2} \) |
| 47 | \( 1 - 5.29T + 47T^{2} \) |
| 53 | \( 1 + 4.14T + 53T^{2} \) |
| 59 | \( 1 + 5.04T + 59T^{2} \) |
| 61 | \( 1 - 1.94T + 61T^{2} \) |
| 67 | \( 1 + 3.87T + 67T^{2} \) |
| 71 | \( 1 + 2.36T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 6.13T + 79T^{2} \) |
| 83 | \( 1 - 7.71T + 83T^{2} \) |
| 89 | \( 1 + 7.59T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.79948318587219426940610692162, −6.99241299866090356405511677084, −6.03969528198650633394118022990, −5.22911130926877174806842976465, −4.75329244120316607137847573767, −3.84343208498299142815748051985, −2.89303291878750414849071301247, −2.18499530510113875722098477582, −1.74955445837761954777054862118, 0,
1.74955445837761954777054862118, 2.18499530510113875722098477582, 2.89303291878750414849071301247, 3.84343208498299142815748051985, 4.75329244120316607137847573767, 5.22911130926877174806842976465, 6.03969528198650633394118022990, 6.99241299866090356405511677084, 7.79948318587219426940610692162
