| L(s) = 1 | + 0.672·2-s + 0.365·3-s − 1.54·4-s + 5-s + 0.245·6-s − 2.74·7-s − 2.38·8-s − 2.86·9-s + 0.672·10-s − 11-s − 0.565·12-s − 1.40·13-s − 1.84·14-s + 0.365·15-s + 1.49·16-s − 7.71·17-s − 1.92·18-s + 6.83·19-s − 1.54·20-s − 1.00·21-s − 0.672·22-s − 1.75·23-s − 0.871·24-s + 25-s − 0.946·26-s − 2.14·27-s + 4.24·28-s + ⋯ |
| L(s) = 1 | + 0.475·2-s + 0.210·3-s − 0.773·4-s + 0.447·5-s + 0.100·6-s − 1.03·7-s − 0.843·8-s − 0.955·9-s + 0.212·10-s − 0.301·11-s − 0.163·12-s − 0.390·13-s − 0.492·14-s + 0.0942·15-s + 0.372·16-s − 1.87·17-s − 0.454·18-s + 1.56·19-s − 0.346·20-s − 0.218·21-s − 0.143·22-s − 0.366·23-s − 0.177·24-s + 0.200·25-s − 0.185·26-s − 0.412·27-s + 0.801·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.150186410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.150186410\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 0.672T + 2T^{2} \) |
| 3 | \( 1 - 0.365T + 3T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 13 | \( 1 + 1.40T + 13T^{2} \) |
| 17 | \( 1 + 7.71T + 17T^{2} \) |
| 19 | \( 1 - 6.83T + 19T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 29 | \( 1 + 5.67T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 2.50T + 37T^{2} \) |
| 41 | \( 1 - 9.29T + 41T^{2} \) |
| 43 | \( 1 + 4.67T + 43T^{2} \) |
| 47 | \( 1 - 5.05T + 47T^{2} \) |
| 53 | \( 1 + 0.00695T + 53T^{2} \) |
| 59 | \( 1 - 4.33T + 59T^{2} \) |
| 61 | \( 1 - 5.28T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 5.36T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 - 0.367T + 97T^{2} \) |
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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
−8.584147252734758796235183218632, −7.80602885083446195106947419512, −6.77212767037657141234583023564, −6.11051386026801563608584905349, −5.45288676861050347630413226759, −4.73109600225241542981055507079, −3.80487877519718939788167131700, −2.98955622836706938529866163299, −2.36036652877315650730863262808, −0.53639786469155853627548669392,
0.53639786469155853627548669392, 2.36036652877315650730863262808, 2.98955622836706938529866163299, 3.80487877519718939788167131700, 4.73109600225241542981055507079, 5.45288676861050347630413226759, 6.11051386026801563608584905349, 6.77212767037657141234583023564, 7.80602885083446195106947419512, 8.584147252734758796235183218632
