| L(s) = 1 | − 2-s + 4-s − 3.44·5-s − 7-s − 8-s + 3.44·10-s − 2·11-s − 4.89·13-s + 14-s + 16-s − 2·17-s + 7.44·19-s − 3.44·20-s + 2·22-s + 23-s + 6.89·25-s + 4.89·26-s − 28-s − 2.89·29-s + 6·31-s − 32-s + 2·34-s + 3.44·35-s − 7.79·37-s − 7.44·38-s + 3.44·40-s + 9.79·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.54·5-s − 0.377·7-s − 0.353·8-s + 1.09·10-s − 0.603·11-s − 1.35·13-s + 0.267·14-s + 0.250·16-s − 0.485·17-s + 1.70·19-s − 0.771·20-s + 0.426·22-s + 0.208·23-s + 1.37·25-s + 0.960·26-s − 0.188·28-s − 0.538·29-s + 1.07·31-s − 0.176·32-s + 0.342·34-s + 0.583·35-s − 1.28·37-s − 1.20·38-s + 0.545·40-s + 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5585449028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5585449028\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 3.44T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 7.79T + 37T^{2} \) |
| 41 | \( 1 - 9.79T + 41T^{2} \) |
| 43 | \( 1 + 2.89T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 3.10T + 67T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 - 2.89T + 73T^{2} \) |
| 79 | \( 1 - 7.89T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 - 7.10T + 89T^{2} \) |
| 97 | \( 1 + 6.89T + 97T^{2} \) |
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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
−9.779248438971436790349631236950, −8.981296378570230157710136272814, −8.050057113882701124054777004681, −7.40958691694476786450167906856, −6.98670064856040608119055429591, −5.54119488052097393706486540617, −4.57458447192399199202565340230, −3.44932887505211656120201533414, −2.53783335702546801626679811217, −0.60053761833786435033192798039,
0.60053761833786435033192798039, 2.53783335702546801626679811217, 3.44932887505211656120201533414, 4.57458447192399199202565340230, 5.54119488052097393706486540617, 6.98670064856040608119055429591, 7.40958691694476786450167906856, 8.050057113882701124054777004681, 8.981296378570230157710136272814, 9.779248438971436790349631236950
