| L(s) = 1 | + 2-s + 1.35·3-s + 4-s + 5-s + 1.35·6-s + 8-s − 1.17·9-s + 10-s − 2.93·11-s + 1.35·12-s − 4.16·13-s + 1.35·15-s + 16-s + 17-s − 1.17·18-s + 2.23·19-s + 20-s − 2.93·22-s + 8.69·23-s + 1.35·24-s + 25-s − 4.16·26-s − 5.64·27-s + 2.53·29-s + 1.35·30-s + 9.15·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.780·3-s + 0.5·4-s + 0.447·5-s + 0.552·6-s + 0.353·8-s − 0.390·9-s + 0.316·10-s − 0.886·11-s + 0.390·12-s − 1.15·13-s + 0.349·15-s + 0.250·16-s + 0.242·17-s − 0.276·18-s + 0.511·19-s + 0.223·20-s − 0.626·22-s + 1.81·23-s + 0.276·24-s + 0.200·25-s − 0.816·26-s − 1.08·27-s + 0.470·29-s + 0.246·30-s + 1.64·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.453448024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.453448024\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.35T + 3T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 13 | \( 1 + 4.16T + 13T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 - 8.69T + 23T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 - 9.15T + 31T^{2} \) |
| 37 | \( 1 + 0.103T + 37T^{2} \) |
| 41 | \( 1 - 4.99T + 41T^{2} \) |
| 43 | \( 1 - 0.552T + 43T^{2} \) |
| 47 | \( 1 - 9.03T + 47T^{2} \) |
| 53 | \( 1 + 0.109T + 53T^{2} \) |
| 59 | \( 1 - 8.41T + 59T^{2} \) |
| 61 | \( 1 + 6.82T + 61T^{2} \) |
| 67 | \( 1 - 3.16T + 67T^{2} \) |
| 71 | \( 1 - 1.49T + 71T^{2} \) |
| 73 | \( 1 + 6.19T + 73T^{2} \) |
| 79 | \( 1 + 6.69T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 1.79T + 89T^{2} \) |
| 97 | \( 1 + 0.714T + 97T^{2} \) |
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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
−7.63515867701301342734785854440, −7.24700214395111408527254574140, −6.35830366611625457003330114184, −5.54194137100907075063701996971, −5.04445655490874860681958500940, −4.36143146822685503476457322233, −3.19454366943067524474965617407, −2.76757199596249954664173452587, −2.23749587277030794224057642173, −0.884248002326481459928752447922,
0.884248002326481459928752447922, 2.23749587277030794224057642173, 2.76757199596249954664173452587, 3.19454366943067524474965617407, 4.36143146822685503476457322233, 5.04445655490874860681958500940, 5.54194137100907075063701996971, 6.35830366611625457003330114184, 7.24700214395111408527254574140, 7.63515867701301342734785854440
