| L(s) = 1 | − 2-s − 3-s + 4-s + 2.41·5-s + 6-s − 0.257·7-s − 8-s + 9-s − 2.41·10-s + 1.56·11-s − 12-s + 13-s + 0.257·14-s − 2.41·15-s + 16-s + 4.26·17-s − 18-s + 1.38·19-s + 2.41·20-s + 0.257·21-s − 1.56·22-s + 8.56·23-s + 24-s + 0.851·25-s − 26-s − 27-s − 0.257·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.08·5-s + 0.408·6-s − 0.0972·7-s − 0.353·8-s + 0.333·9-s − 0.764·10-s + 0.473·11-s − 0.288·12-s + 0.277·13-s + 0.0687·14-s − 0.624·15-s + 0.250·16-s + 1.03·17-s − 0.235·18-s + 0.317·19-s + 0.540·20-s + 0.0561·21-s − 0.334·22-s + 1.78·23-s + 0.204·24-s + 0.170·25-s − 0.196·26-s − 0.192·27-s − 0.0486·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.926417171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.926417171\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 2.41T + 5T^{2} \) |
| 7 | \( 1 + 0.257T + 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 17 | \( 1 - 4.26T + 17T^{2} \) |
| 19 | \( 1 - 1.38T + 19T^{2} \) |
| 23 | \( 1 - 8.56T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 6.51T + 43T^{2} \) |
| 47 | \( 1 - 9.53T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 4.00T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 9.28T + 67T^{2} \) |
| 71 | \( 1 + 0.286T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 - 9.06T + 79T^{2} \) |
| 83 | \( 1 + 6.84T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−7.900133359114379908315264562145, −6.95466494170447354731839515064, −6.52944036502347442566351860695, −5.88258793442047478118191418969, −5.22690695539968134372981154372, −4.46653313279949315688766266207, −3.24289333867952388407368570604, −2.59089285916287069328447111170, −1.37754656900372574842551980265, −0.918449103972021885604456540543,
0.918449103972021885604456540543, 1.37754656900372574842551980265, 2.59089285916287069328447111170, 3.24289333867952388407368570604, 4.46653313279949315688766266207, 5.22690695539968134372981154372, 5.88258793442047478118191418969, 6.52944036502347442566351860695, 6.95466494170447354731839515064, 7.900133359114379908315264562145
