| L(s) = 1 | − 2-s − 0.732·3-s + 4-s − 5-s + 0.732·6-s + 7-s − 8-s − 2.46·9-s + 10-s − 0.732·12-s + 4·13-s − 14-s + 0.732·15-s + 16-s + 7.46·17-s + 2.46·18-s + 2.73·19-s − 20-s − 0.732·21-s − 2.19·23-s + 0.732·24-s + 25-s − 4·26-s + 4·27-s + 28-s + 9.66·29-s − 0.732·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.422·3-s + 0.5·4-s − 0.447·5-s + 0.298·6-s + 0.377·7-s − 0.353·8-s − 0.821·9-s + 0.316·10-s − 0.211·12-s + 1.10·13-s − 0.267·14-s + 0.189·15-s + 0.250·16-s + 1.81·17-s + 0.580·18-s + 0.626·19-s − 0.223·20-s − 0.159·21-s − 0.457·23-s + 0.149·24-s + 0.200·25-s − 0.784·26-s + 0.769·27-s + 0.188·28-s + 1.79·29-s − 0.133·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.356778906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.356778906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 - 2.73T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 - 9.66T + 29T^{2} \) |
| 31 | \( 1 - 7.46T + 31T^{2} \) |
| 37 | \( 1 - 0.732T + 37T^{2} \) |
| 41 | \( 1 + 8.19T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 8.73T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 9.66T + 79T^{2} \) |
| 83 | \( 1 - 1.07T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 + 0.732T + 97T^{2} \) |
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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
−7.948602980300326526203846525177, −7.27458211560247622726114967852, −6.31944989851239131294642289650, −5.88869373553919900668073849168, −5.15990683524937834056175508296, −4.26330379660979631485177717867, −3.26931317018699918414215610111, −2.75469352950796797406717442898, −1.33382075137780746362590914466, −0.74346325241763926075969472272,
0.74346325241763926075969472272, 1.33382075137780746362590914466, 2.75469352950796797406717442898, 3.26931317018699918414215610111, 4.26330379660979631485177717867, 5.15990683524937834056175508296, 5.88869373553919900668073849168, 6.31944989851239131294642289650, 7.27458211560247622726114967852, 7.948602980300326526203846525177
