| L(s) = 1 | − 1.14·2-s − 3-s − 0.680·4-s + 1.14·6-s − 3.52·7-s + 3.07·8-s + 9-s + 5.92·11-s + 0.680·12-s − 2.69·13-s + 4.04·14-s − 2.17·16-s + 5.21·17-s − 1.14·18-s − 4.16·19-s + 3.52·21-s − 6.80·22-s − 3.38·23-s − 3.07·24-s + 3.09·26-s − 27-s + 2.39·28-s − 29-s + 1.42·31-s − 3.65·32-s − 5.92·33-s − 5.99·34-s + ⋯ |
| L(s) = 1 | − 0.812·2-s − 0.577·3-s − 0.340·4-s + 0.469·6-s − 1.33·7-s + 1.08·8-s + 0.333·9-s + 1.78·11-s + 0.196·12-s − 0.746·13-s + 1.08·14-s − 0.544·16-s + 1.26·17-s − 0.270·18-s − 0.956·19-s + 0.768·21-s − 1.45·22-s − 0.705·23-s − 0.628·24-s + 0.606·26-s − 0.192·27-s + 0.452·28-s − 0.185·29-s + 0.255·31-s − 0.646·32-s − 1.03·33-s − 1.02·34-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
| \(L(1)\) |
\(\approx\) |
\(0.5646255482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5646255482\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.14T + 2T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 11 | \( 1 - 5.92T + 11T^{2} \) |
| 13 | \( 1 + 2.69T + 13T^{2} \) |
| 17 | \( 1 - 5.21T + 17T^{2} \) |
| 19 | \( 1 + 4.16T + 19T^{2} \) |
| 23 | \( 1 + 3.38T + 23T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 + 3.36T + 37T^{2} \) |
| 41 | \( 1 - 3.78T + 41T^{2} \) |
| 43 | \( 1 + 4.17T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 6.14T + 53T^{2} \) |
| 59 | \( 1 + 2.59T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 8.72T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 5.58T + 73T^{2} \) |
| 79 | \( 1 - 5.07T + 79T^{2} \) |
| 83 | \( 1 - 1.68T + 83T^{2} \) |
| 89 | \( 1 - 3.16T + 89T^{2} \) |
| 97 | \( 1 - 6.72T + 97T^{2} \) |
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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
−9.318457848571595702208551918497, −8.432452131819620144158406363708, −7.54675591590105769616006329080, −6.66753181040984061777160331774, −6.21421326042430814936255942425, −5.09993654205724016729633550915, −4.11307252849259335785251141687, −3.41867480863150989055727545481, −1.78161934335717031107631019572, −0.58370932640766198592224533619,
0.58370932640766198592224533619, 1.78161934335717031107631019572, 3.41867480863150989055727545481, 4.11307252849259335785251141687, 5.09993654205724016729633550915, 6.21421326042430814936255942425, 6.66753181040984061777160331774, 7.54675591590105769616006329080, 8.432452131819620144158406363708, 9.318457848571595702208551918497
