| L(s) = 1 | + 1.32·3-s − 0.265·5-s + 7-s − 1.25·9-s + 11-s − 13-s − 0.350·15-s + 6.26·17-s − 0.906·19-s + 1.32·21-s + 2.91·23-s − 4.92·25-s − 5.62·27-s + 9.05·29-s − 1.90·31-s + 1.32·33-s − 0.265·35-s − 4.42·37-s − 1.32·39-s + 10.9·41-s + 11.7·43-s + 0.331·45-s − 7.57·47-s + 49-s + 8.29·51-s − 7.67·53-s − 0.265·55-s + ⋯ |
| L(s) = 1 | + 0.763·3-s − 0.118·5-s + 0.377·7-s − 0.416·9-s + 0.301·11-s − 0.277·13-s − 0.0905·15-s + 1.52·17-s − 0.207·19-s + 0.288·21-s + 0.608·23-s − 0.985·25-s − 1.08·27-s + 1.68·29-s − 0.341·31-s + 0.230·33-s − 0.0448·35-s − 0.728·37-s − 0.211·39-s + 1.71·41-s + 1.79·43-s + 0.0493·45-s − 1.10·47-s + 0.142·49-s + 1.16·51-s − 1.05·53-s − 0.0357·55-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
| \(L(1)\) |
\(\approx\) |
\(2.580752536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.580752536\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.32T + 3T^{2} \) |
| 5 | \( 1 + 0.265T + 5T^{2} \) |
| 17 | \( 1 - 6.26T + 17T^{2} \) |
| 19 | \( 1 + 0.906T + 19T^{2} \) |
| 23 | \( 1 - 2.91T + 23T^{2} \) |
| 29 | \( 1 - 9.05T + 29T^{2} \) |
| 31 | \( 1 + 1.90T + 31T^{2} \) |
| 37 | \( 1 + 4.42T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 7.57T + 47T^{2} \) |
| 53 | \( 1 + 7.67T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 5.49T + 67T^{2} \) |
| 71 | \( 1 - 6.99T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 5.79T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 7.68T + 97T^{2} \) |
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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
−8.395839971854792482193368123580, −7.80848412214145984601798614807, −7.24539359875057412069125537328, −6.16207428016317958129192510745, −5.51097296737178670319005873976, −4.60366326030871274644054454344, −3.69975688695487909064380092163, −2.97483367150130843695231480684, −2.11856274965806966026566009313, −0.907078862581200667301849797417,
0.907078862581200667301849797417, 2.11856274965806966026566009313, 2.97483367150130843695231480684, 3.69975688695487909064380092163, 4.60366326030871274644054454344, 5.51097296737178670319005873976, 6.16207428016317958129192510745, 7.24539359875057412069125537328, 7.80848412214145984601798614807, 8.395839971854792482193368123580
