| L(s) = 1 | + 0.792·2-s + 3-s − 1.37·4-s + 0.792·6-s − 2.01·7-s − 2.67·8-s + 9-s + 2.65·11-s − 1.37·12-s − 2.48·13-s − 1.60·14-s + 0.624·16-s + 7.14·17-s + 0.792·18-s + 5.22·19-s − 2.01·21-s + 2.10·22-s − 8.36·23-s − 2.67·24-s − 1.96·26-s + 27-s + 2.76·28-s − 29-s + 2.40·31-s + 5.84·32-s + 2.65·33-s + 5.66·34-s + ⋯ |
| L(s) = 1 | + 0.560·2-s + 0.577·3-s − 0.685·4-s + 0.323·6-s − 0.762·7-s − 0.944·8-s + 0.333·9-s + 0.800·11-s − 0.395·12-s − 0.688·13-s − 0.427·14-s + 0.156·16-s + 1.73·17-s + 0.186·18-s + 1.19·19-s − 0.440·21-s + 0.448·22-s − 1.74·23-s − 0.545·24-s − 0.386·26-s + 0.192·27-s + 0.523·28-s − 0.185·29-s + 0.431·31-s + 1.03·32-s + 0.461·33-s + 0.971·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.210339115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.210339115\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.792T + 2T^{2} \) |
| 7 | \( 1 + 2.01T + 7T^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 13 | \( 1 + 2.48T + 13T^{2} \) |
| 17 | \( 1 - 7.14T + 17T^{2} \) |
| 19 | \( 1 - 5.22T + 19T^{2} \) |
| 23 | \( 1 + 8.36T + 23T^{2} \) |
| 31 | \( 1 - 2.40T + 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 - 4.74T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 0.566T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 7.06T + 89T^{2} \) |
| 97 | \( 1 + 5.34T + 97T^{2} \) |
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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
−9.329879124813399544264770496692, −8.193206493513634592817344734323, −7.68782517953278331249356467068, −6.61239371938253270449840263725, −5.79137093009086503043083161272, −5.06492921450382682837872790256, −3.90001562621694127418893434395, −3.53664599290142752292649616269, −2.51006271903330117460783105782, −0.893926187061001652031481912570,
0.893926187061001652031481912570, 2.51006271903330117460783105782, 3.53664599290142752292649616269, 3.90001562621694127418893434395, 5.06492921450382682837872790256, 5.79137093009086503043083161272, 6.61239371938253270449840263725, 7.68782517953278331249356467068, 8.193206493513634592817344734323, 9.329879124813399544264770496692
