| L(s) = 1 | + 2-s + 3-s + 4-s − 2.82·5-s + 6-s + 8-s + 9-s − 2.82·10-s + 2·11-s + 12-s + 2.82·13-s − 2.82·15-s + 16-s + 18-s − 19-s − 2.82·20-s + 2·22-s + 4.82·23-s + 24-s + 3.00·25-s + 2.82·26-s + 27-s − 3.65·29-s − 2.82·30-s − 1.17·31-s + 32-s + 2·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.26·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.894·10-s + 0.603·11-s + 0.288·12-s + 0.784·13-s − 0.730·15-s + 0.250·16-s + 0.235·18-s − 0.229·19-s − 0.632·20-s + 0.426·22-s + 1.00·23-s + 0.204·24-s + 0.600·25-s + 0.554·26-s + 0.192·27-s − 0.679·29-s − 0.516·30-s − 0.210·31-s + 0.176·32-s + 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.375464919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.375464919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 6.48T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 - 1.65T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 8.82T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 7.31T + 97T^{2} \) |
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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
−7.982784846887823407557650211915, −7.44930798068353424461146170199, −6.80379265596786717078071620695, −6.02118053957939202579059258839, −5.07747549851032221009240718688, −4.21938683447945886842154960544, −3.74025073656473530275038430221, −3.14852122178205003756549848141, −2.05570968753705216845594622779, −0.877555706878572268348603880081,
0.877555706878572268348603880081, 2.05570968753705216845594622779, 3.14852122178205003756549848141, 3.74025073656473530275038430221, 4.21938683447945886842154960544, 5.07747549851032221009240718688, 6.02118053957939202579059258839, 6.80379265596786717078071620695, 7.44930798068353424461146170199, 7.982784846887823407557650211915
