L(s) = 1 | + 0.592·2-s + 3-s − 1.64·4-s + 5-s + 0.592·6-s − 0.542·7-s − 2.16·8-s + 9-s + 0.592·10-s − 2.37·11-s − 1.64·12-s + 4.21·13-s − 0.321·14-s + 15-s + 2.01·16-s − 4.91·17-s + 0.592·18-s + 0.967·19-s − 1.64·20-s − 0.542·21-s − 1.40·22-s − 8.66·23-s − 2.16·24-s + 25-s + 2.49·26-s + 27-s + 0.895·28-s + ⋯ |
L(s) = 1 | + 0.418·2-s + 0.577·3-s − 0.824·4-s + 0.447·5-s + 0.241·6-s − 0.205·7-s − 0.764·8-s + 0.333·9-s + 0.187·10-s − 0.714·11-s − 0.476·12-s + 1.16·13-s − 0.0859·14-s + 0.258·15-s + 0.504·16-s − 1.19·17-s + 0.139·18-s + 0.222·19-s − 0.368·20-s − 0.118·21-s − 0.299·22-s − 1.80·23-s − 0.441·24-s + 0.200·25-s + 0.489·26-s + 0.192·27-s + 0.169·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 - 0.592T + 2T^{2} \) |
| 7 | \( 1 + 0.542T + 7T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 13 | \( 1 - 4.21T + 13T^{2} \) |
| 17 | \( 1 + 4.91T + 17T^{2} \) |
| 19 | \( 1 - 0.967T + 19T^{2} \) |
| 23 | \( 1 + 8.66T + 23T^{2} \) |
| 29 | \( 1 - 9.93T + 29T^{2} \) |
| 31 | \( 1 - 2.19T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 0.524T + 41T^{2} \) |
| 43 | \( 1 - 3.25T + 43T^{2} \) |
| 47 | \( 1 + 0.431T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 4.15T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 3.38T + 67T^{2} \) |
| 71 | \( 1 + 8.00T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 5.64T + 83T^{2} \) |
| 89 | \( 1 + 6.69T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−7.990030780703438114799973849981, −6.89195709342035838394425402348, −6.17731862048978842372795631135, −5.59949775888224976816428184689, −4.64908223304161057272644735434, −4.12708871461947669356703702400, −3.26385845303818861223801342806, −2.53029530663843365348907313286, −1.42855125426562013488839771125, 0,
1.42855125426562013488839771125, 2.53029530663843365348907313286, 3.26385845303818861223801342806, 4.12708871461947669356703702400, 4.64908223304161057272644735434, 5.59949775888224976816428184689, 6.17731862048978842372795631135, 6.89195709342035838394425402348, 7.990030780703438114799973849981