L(s) = 1 | − 1.95·3-s − 0.209·5-s − 7-s + 0.827·9-s + 11-s − 13-s + 0.408·15-s + 2.88·17-s − 1.89·19-s + 1.95·21-s − 1.33·23-s − 4.95·25-s + 4.25·27-s + 5.48·29-s + 4.65·31-s − 1.95·33-s + 0.209·35-s + 1.56·37-s + 1.95·39-s − 7.25·41-s + 4.32·43-s − 0.172·45-s + 3.41·47-s + 49-s − 5.64·51-s − 10.2·53-s − 0.209·55-s + ⋯ |
L(s) = 1 | − 1.12·3-s − 0.0934·5-s − 0.377·7-s + 0.275·9-s + 0.301·11-s − 0.277·13-s + 0.105·15-s + 0.699·17-s − 0.435·19-s + 0.426·21-s − 0.279·23-s − 0.991·25-s + 0.818·27-s + 1.01·29-s + 0.835·31-s − 0.340·33-s + 0.0353·35-s + 0.258·37-s + 0.313·39-s − 1.13·41-s + 0.659·43-s − 0.0257·45-s + 0.497·47-s + 0.142·49-s − 0.790·51-s − 1.40·53-s − 0.0281·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 + 0.209T + 5T^{2} \) |
| 17 | \( 1 - 2.88T + 17T^{2} \) |
| 19 | \( 1 + 1.89T + 19T^{2} \) |
| 23 | \( 1 + 1.33T + 23T^{2} \) |
| 29 | \( 1 - 5.48T + 29T^{2} \) |
| 31 | \( 1 - 4.65T + 31T^{2} \) |
| 37 | \( 1 - 1.56T + 37T^{2} \) |
| 41 | \( 1 + 7.25T + 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 - 3.41T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 8.91T + 59T^{2} \) |
| 61 | \( 1 + 0.657T + 61T^{2} \) |
| 67 | \( 1 - 2.30T + 67T^{2} \) |
| 71 | \( 1 + 8.47T + 71T^{2} \) |
| 73 | \( 1 + 4.90T + 73T^{2} \) |
| 79 | \( 1 - 7.52T + 79T^{2} \) |
| 83 | \( 1 - 1.60T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−8.055966220084498617764817631956, −7.21519324940920901095019987591, −6.37920073912689053281199461394, −6.00263703759491526335786276763, −5.14503444369706779597684160915, −4.43525450851473754373342687096, −3.49227547180444923993031098197, −2.47490101735427255017314982929, −1.13940660503600657452161457975, 0,
1.13940660503600657452161457975, 2.47490101735427255017314982929, 3.49227547180444923993031098197, 4.43525450851473754373342687096, 5.14503444369706779597684160915, 6.00263703759491526335786276763, 6.37920073912689053281199461394, 7.21519324940920901095019987591, 8.055966220084498617764817631956