| L(s) = 1 | + 3.29·3-s − 1.31·5-s + 2.99·7-s + 7.86·9-s − 0.871·11-s + 5.19·13-s − 4.33·15-s + 3.90·17-s − 1.65·19-s + 9.88·21-s − 5.64·23-s − 3.26·25-s + 16.0·27-s + 5.33·29-s − 9.19·31-s − 2.87·33-s − 3.94·35-s + 8.95·37-s + 17.1·39-s + 3.01·41-s − 9.78·43-s − 10.3·45-s + 4.72·47-s + 1.99·49-s + 12.8·51-s + 7.33·53-s + 1.14·55-s + ⋯ |
| L(s) = 1 | + 1.90·3-s − 0.588·5-s + 1.13·7-s + 2.62·9-s − 0.262·11-s + 1.44·13-s − 1.12·15-s + 0.946·17-s − 0.380·19-s + 2.15·21-s − 1.17·23-s − 0.653·25-s + 3.08·27-s + 0.990·29-s − 1.65·31-s − 0.500·33-s − 0.667·35-s + 1.47·37-s + 2.74·39-s + 0.470·41-s − 1.49·43-s − 1.54·45-s + 0.689·47-s + 0.284·49-s + 1.80·51-s + 1.00·53-s + 0.154·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.497816052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.497816052\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 3.29T + 3T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 - 2.99T + 7T^{2} \) |
| 11 | \( 1 + 0.871T + 11T^{2} \) |
| 13 | \( 1 - 5.19T + 13T^{2} \) |
| 17 | \( 1 - 3.90T + 17T^{2} \) |
| 19 | \( 1 + 1.65T + 19T^{2} \) |
| 23 | \( 1 + 5.64T + 23T^{2} \) |
| 29 | \( 1 - 5.33T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 - 8.95T + 37T^{2} \) |
| 41 | \( 1 - 3.01T + 41T^{2} \) |
| 43 | \( 1 + 9.78T + 43T^{2} \) |
| 47 | \( 1 - 4.72T + 47T^{2} \) |
| 53 | \( 1 - 7.33T + 53T^{2} \) |
| 59 | \( 1 + 7.72T + 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 8.42T + 71T^{2} \) |
| 73 | \( 1 - 8.67T + 73T^{2} \) |
| 79 | \( 1 + 4.92T + 79T^{2} \) |
| 83 | \( 1 - 1.80T + 83T^{2} \) |
| 89 | \( 1 - 9.34T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.207148699154853552150789966297, −8.003064610755308745961088029023, −7.46540519385299725662711774762, −6.39862055231979881095031331335, −5.33374532111888636663467811728, −4.15449867712991776009232339699, −3.91818748884236864814425401093, −3.00477512740298360628762132298, −2.01672353158102721790764034996, −1.27333502765704506729985154822,
1.27333502765704506729985154822, 2.01672353158102721790764034996, 3.00477512740298360628762132298, 3.91818748884236864814425401093, 4.15449867712991776009232339699, 5.33374532111888636663467811728, 6.39862055231979881095031331335, 7.46540519385299725662711774762, 8.003064610755308745961088029023, 8.207148699154853552150789966297
