| L(s) = 1 | + 3.30·3-s − 1.14·5-s − 7-s + 7.95·9-s − 11-s − 13-s − 3.77·15-s + 6.91·17-s − 0.199·19-s − 3.30·21-s + 3.36·23-s − 3.69·25-s + 16.3·27-s + 2.35·29-s − 2.74·31-s − 3.30·33-s + 1.14·35-s + 5.34·37-s − 3.30·39-s + 8.92·41-s − 1.44·43-s − 9.07·45-s − 0.453·47-s + 49-s + 22.8·51-s − 4.92·53-s + 1.14·55-s + ⋯ |
| L(s) = 1 | + 1.91·3-s − 0.510·5-s − 0.377·7-s + 2.65·9-s − 0.301·11-s − 0.277·13-s − 0.975·15-s + 1.67·17-s − 0.0456·19-s − 0.722·21-s + 0.700·23-s − 0.739·25-s + 3.15·27-s + 0.436·29-s − 0.492·31-s − 0.576·33-s + 0.192·35-s + 0.878·37-s − 0.529·39-s + 1.39·41-s − 0.220·43-s − 1.35·45-s − 0.0661·47-s + 0.142·49-s + 3.20·51-s − 0.675·53-s + 0.153·55-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
| \(L(1)\) |
\(\approx\) |
\(3.737658800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.737658800\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 + 1.14T + 5T^{2} \) |
| 17 | \( 1 - 6.91T + 17T^{2} \) |
| 19 | \( 1 + 0.199T + 19T^{2} \) |
| 23 | \( 1 - 3.36T + 23T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 - 5.34T + 37T^{2} \) |
| 41 | \( 1 - 8.92T + 41T^{2} \) |
| 43 | \( 1 + 1.44T + 43T^{2} \) |
| 47 | \( 1 + 0.453T + 47T^{2} \) |
| 53 | \( 1 + 4.92T + 53T^{2} \) |
| 59 | \( 1 + 2.39T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 1.28T + 67T^{2} \) |
| 71 | \( 1 + 7.69T + 71T^{2} \) |
| 73 | \( 1 - 9.77T + 73T^{2} \) |
| 79 | \( 1 - 8.24T + 79T^{2} \) |
| 83 | \( 1 - 1.96T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−8.327411721143866225837635727088, −7.73128464549458590502823601982, −7.45495871361793018815426661101, −6.48171960989691645622460473360, −5.34125027894193229549236764925, −4.33849820526570018865070105929, −3.61424844345391080088174793718, −3.03099715057741918116386375026, −2.26179818917661492741910081567, −1.06693675795323359981674547799,
1.06693675795323359981674547799, 2.26179818917661492741910081567, 3.03099715057741918116386375026, 3.61424844345391080088174793718, 4.33849820526570018865070105929, 5.34125027894193229549236764925, 6.48171960989691645622460473360, 7.45495871361793018815426661101, 7.73128464549458590502823601982, 8.327411721143866225837635727088
