L(s) = 1 | − 2.14·2-s + 3-s + 2.62·4-s + 3.73·5-s − 2.14·6-s − 1.33·8-s + 9-s − 8.01·10-s + 0.528·11-s + 2.62·12-s + 13-s + 3.73·15-s − 2.37·16-s + 3.80·17-s − 2.14·18-s − 3.88·19-s + 9.77·20-s − 1.13·22-s − 2.43·23-s − 1.33·24-s + 8.91·25-s − 2.14·26-s + 27-s − 3.56·29-s − 8.01·30-s + 4.99·31-s + 7.77·32-s + ⋯ |
L(s) = 1 | − 1.51·2-s + 0.577·3-s + 1.31·4-s + 1.66·5-s − 0.877·6-s − 0.471·8-s + 0.333·9-s − 2.53·10-s + 0.159·11-s + 0.756·12-s + 0.277·13-s + 0.963·15-s − 0.593·16-s + 0.922·17-s − 0.506·18-s − 0.890·19-s + 2.18·20-s − 0.242·22-s − 0.507·23-s − 0.272·24-s + 1.78·25-s − 0.421·26-s + 0.192·27-s − 0.662·29-s − 1.46·30-s + 0.896·31-s + 1.37·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.488701290\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.488701290\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 5 | \( 1 - 3.73T + 5T^{2} \) |
| 11 | \( 1 - 0.528T + 11T^{2} \) |
| 17 | \( 1 - 3.80T + 17T^{2} \) |
| 19 | \( 1 + 3.88T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 3.56T + 29T^{2} \) |
| 31 | \( 1 - 4.99T + 31T^{2} \) |
| 37 | \( 1 - 9.45T + 37T^{2} \) |
| 41 | \( 1 + 5.21T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 7.62T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 5.80T + 61T^{2} \) |
| 67 | \( 1 - 6.15T + 67T^{2} \) |
| 71 | \( 1 + 6.88T + 71T^{2} \) |
| 73 | \( 1 + 6.45T + 73T^{2} \) |
| 79 | \( 1 + 3.70T + 79T^{2} \) |
| 83 | \( 1 + 2.16T + 83T^{2} \) |
| 89 | \( 1 - 1.46T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−9.325672978458200227013277938952, −8.534632434845777403864573550882, −8.001040957293870417696761250105, −6.98612068643774123489354673255, −6.28225967683316460161252175814, −5.46867620700542721019467214194, −4.17941236448272417310710874317, −2.71567289116766517503981629000, −1.97685858521331521560848807983, −1.08228552817694683248334058361,
1.08228552817694683248334058361, 1.97685858521331521560848807983, 2.71567289116766517503981629000, 4.17941236448272417310710874317, 5.46867620700542721019467214194, 6.28225967683316460161252175814, 6.98612068643774123489354673255, 8.001040957293870417696761250105, 8.534632434845777403864573550882, 9.325672978458200227013277938952