| L(s) = 1 | + 0.449·2-s − 1.79·4-s − 3.08·5-s − 7-s − 1.70·8-s − 1.38·10-s + 2.74·11-s − 6.63·13-s − 0.449·14-s + 2.82·16-s + 17-s + 4.90·19-s + 5.54·20-s + 1.23·22-s + 4.79·23-s + 4.51·25-s − 2.98·26-s + 1.79·28-s + 7.72·29-s − 2.28·31-s + 4.68·32-s + 0.449·34-s + 3.08·35-s − 5.05·37-s + 2.20·38-s + 5.27·40-s + 2.96·41-s + ⋯ |
| L(s) = 1 | + 0.318·2-s − 0.898·4-s − 1.37·5-s − 0.377·7-s − 0.604·8-s − 0.438·10-s + 0.828·11-s − 1.83·13-s − 0.120·14-s + 0.706·16-s + 0.242·17-s + 1.12·19-s + 1.24·20-s + 0.263·22-s + 1.00·23-s + 0.903·25-s − 0.585·26-s + 0.339·28-s + 1.43·29-s − 0.410·31-s + 0.828·32-s + 0.0771·34-s + 0.521·35-s − 0.831·37-s + 0.358·38-s + 0.833·40-s + 0.462·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8899132656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8899132656\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 0.449T + 2T^{2} \) |
| 5 | \( 1 + 3.08T + 5T^{2} \) |
| 11 | \( 1 - 2.74T + 11T^{2} \) |
| 13 | \( 1 + 6.63T + 13T^{2} \) |
| 19 | \( 1 - 4.90T + 19T^{2} \) |
| 23 | \( 1 - 4.79T + 23T^{2} \) |
| 29 | \( 1 - 7.72T + 29T^{2} \) |
| 31 | \( 1 + 2.28T + 31T^{2} \) |
| 37 | \( 1 + 5.05T + 37T^{2} \) |
| 41 | \( 1 - 2.96T + 41T^{2} \) |
| 43 | \( 1 - 8.71T + 43T^{2} \) |
| 47 | \( 1 + 5.00T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 5.90T + 61T^{2} \) |
| 67 | \( 1 + 4.06T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 3.67T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−9.623088372090789126442657085329, −9.224466278206910742839580686001, −8.150246741265250371074278317999, −7.46277525793650017023936060449, −6.65466316554051010785564905513, −5.25466509674424402745442191111, −4.64045428812828744178940289223, −3.72466879945049816783792676005, −2.93557156275318341908028706256, −0.68502657694020328366558904489,
0.68502657694020328366558904489, 2.93557156275318341908028706256, 3.72466879945049816783792676005, 4.64045428812828744178940289223, 5.25466509674424402745442191111, 6.65466316554051010785564905513, 7.46277525793650017023936060449, 8.150246741265250371074278317999, 9.224466278206910742839580686001, 9.623088372090789126442657085329
