| L(s) = 1 | + 3·3-s − 26.7·7-s + 9·9-s + 46.6·11-s − 49.6·13-s − 32.4·17-s + 4.14·19-s − 80.2·21-s + 213.·23-s + 27·27-s − 272.·29-s − 327.·31-s + 139.·33-s + 399.·37-s − 148.·39-s − 42.2·41-s + 468.·43-s − 275.·47-s + 372.·49-s − 97.2·51-s − 158.·53-s + 12.4·57-s − 566.·59-s + 206.·61-s − 240.·63-s − 351.·67-s + 639.·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.44·7-s + 0.333·9-s + 1.27·11-s − 1.05·13-s − 0.462·17-s + 0.0500·19-s − 0.833·21-s + 1.93·23-s + 0.192·27-s − 1.74·29-s − 1.89·31-s + 0.737·33-s + 1.77·37-s − 0.610·39-s − 0.161·41-s + 1.66·43-s − 0.853·47-s + 1.08·49-s − 0.266·51-s − 0.409·53-s + 0.0288·57-s − 1.24·59-s + 0.433·61-s − 0.481·63-s − 0.641·67-s + 1.11·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.001838721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.001838721\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 26.7T + 343T^{2} \) |
| 11 | \( 1 - 46.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 32.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.14T + 6.85e3T^{2} \) |
| 23 | \( 1 - 213.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 272.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 327.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 399.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 42.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 468.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 275.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 158.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 566.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 206.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 351.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 500.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 987.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 172.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 238.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 42.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.18e3T + 9.12e5T^{2} \) |
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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
−9.027552346133569878804130264420, −7.65783367435389114721481783467, −7.09551633655438119671592543917, −6.46871214446932044957076658271, −5.55732854306828055711944716413, −4.44489329802607383264972511796, −3.60825473306669661635260547063, −2.93526051762979582437687522538, −1.90663345487212514898696022225, −0.59417055895269783585569381852,
0.59417055895269783585569381852, 1.90663345487212514898696022225, 2.93526051762979582437687522538, 3.60825473306669661635260547063, 4.44489329802607383264972511796, 5.55732854306828055711944716413, 6.46871214446932044957076658271, 7.09551633655438119671592543917, 7.65783367435389114721481783467, 9.027552346133569878804130264420
