| L(s) = 1 | + 1.20·2-s + 2.19·3-s − 0.543·4-s − 2.54·5-s + 2.64·6-s + 1.13·7-s − 3.06·8-s + 1.81·9-s − 3.07·10-s + 5.16·11-s − 1.19·12-s + 2.58·13-s + 1.37·14-s − 5.58·15-s − 2.61·16-s − 3.09·17-s + 2.18·18-s + 8.23·19-s + 1.38·20-s + 2.50·21-s + 6.23·22-s + 3.81·23-s − 6.73·24-s + 1.47·25-s + 3.11·26-s − 2.60·27-s − 0.619·28-s + ⋯ |
| L(s) = 1 | + 0.853·2-s + 1.26·3-s − 0.271·4-s − 1.13·5-s + 1.08·6-s + 0.430·7-s − 1.08·8-s + 0.604·9-s − 0.971·10-s + 1.55·11-s − 0.344·12-s + 0.716·13-s + 0.367·14-s − 1.44·15-s − 0.654·16-s − 0.750·17-s + 0.516·18-s + 1.89·19-s + 0.309·20-s + 0.545·21-s + 1.32·22-s + 0.795·23-s − 1.37·24-s + 0.295·25-s + 0.611·26-s − 0.500·27-s − 0.117·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.310451963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.310451963\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 \) |
| good | 2 | \( 1 - 1.20T + 2T^{2} \) |
| 3 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 + 2.54T + 5T^{2} \) |
| 7 | \( 1 - 1.13T + 7T^{2} \) |
| 11 | \( 1 - 5.16T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 19 | \( 1 - 8.23T + 19T^{2} \) |
| 23 | \( 1 - 3.81T + 23T^{2} \) |
| 29 | \( 1 - 5.79T + 29T^{2} \) |
| 31 | \( 1 - 4.51T + 31T^{2} \) |
| 37 | \( 1 + 0.722T + 37T^{2} \) |
| 43 | \( 1 - 1.84T + 43T^{2} \) |
| 47 | \( 1 - 3.54T + 47T^{2} \) |
| 53 | \( 1 - 5.59T + 53T^{2} \) |
| 59 | \( 1 - 0.785T + 59T^{2} \) |
| 61 | \( 1 + 7.05T + 61T^{2} \) |
| 67 | \( 1 + 5.34T + 67T^{2} \) |
| 71 | \( 1 - 4.53T + 71T^{2} \) |
| 73 | \( 1 + 0.149T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 - 4.27T + 83T^{2} \) |
| 89 | \( 1 - 1.70T + 89T^{2} \) |
| 97 | \( 1 + 4.39T + 97T^{2} \) |
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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
−8.985701676546895230998915339765, −8.709171623855238645004232421486, −7.88683171282843165605578490881, −7.05250500501462226743592324597, −6.08672965371974970277867103865, −4.87088051304442286264034933674, −4.12065756391916514969741085817, −3.53591576033733856423938366684, −2.84867985152134778835706651353, −1.15360620689245417388010593238,
1.15360620689245417388010593238, 2.84867985152134778835706651353, 3.53591576033733856423938366684, 4.12065756391916514969741085817, 4.87088051304442286264034933674, 6.08672965371974970277867103865, 7.05250500501462226743592324597, 7.88683171282843165605578490881, 8.709171623855238645004232421486, 8.985701676546895230998915339765
