| L(s) = 1 | + 1.21·2-s + 1.77·3-s − 0.529·4-s − 5-s + 2.14·6-s − 3.37·7-s − 3.06·8-s + 0.142·9-s − 1.21·10-s − 11-s − 0.939·12-s − 0.439·13-s − 4.08·14-s − 1.77·15-s − 2.65·16-s − 3.08·17-s + 0.172·18-s − 19-s + 0.529·20-s − 5.97·21-s − 1.21·22-s + 1.45·23-s − 5.43·24-s + 25-s − 0.533·26-s − 5.06·27-s + 1.78·28-s + ⋯ |
| L(s) = 1 | + 0.857·2-s + 1.02·3-s − 0.264·4-s − 0.447·5-s + 0.877·6-s − 1.27·7-s − 1.08·8-s + 0.0474·9-s − 0.383·10-s − 0.301·11-s − 0.271·12-s − 0.121·13-s − 1.09·14-s − 0.457·15-s − 0.664·16-s − 0.749·17-s + 0.0406·18-s − 0.229·19-s + 0.118·20-s − 1.30·21-s − 0.258·22-s + 0.303·23-s − 1.10·24-s + 0.200·25-s − 0.104·26-s − 0.974·27-s + 0.337·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 3 | \( 1 - 1.77T + 3T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 13 | \( 1 + 0.439T + 13T^{2} \) |
| 17 | \( 1 + 3.08T + 17T^{2} \) |
| 23 | \( 1 - 1.45T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 + 6.80T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 - 4.59T + 43T^{2} \) |
| 47 | \( 1 - 0.210T + 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 5.66T + 61T^{2} \) |
| 67 | \( 1 - 6.14T + 67T^{2} \) |
| 71 | \( 1 + 7.15T + 71T^{2} \) |
| 73 | \( 1 - 4.15T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 3.35T + 83T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 + 6.70T + 97T^{2} \) |
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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
−9.210986761182075897454567073949, −8.882289782398650893899368565683, −7.931702908710211417573937074151, −6.87581691905059560806756997204, −6.04658597204777017181920780725, −4.97621715822582253899070581847, −3.93438028005217309523548015244, −3.26928278290641127738780745165, −2.50851785700766887540804232986, 0,
2.50851785700766887540804232986, 3.26928278290641127738780745165, 3.93438028005217309523548015244, 4.97621715822582253899070581847, 6.04658597204777017181920780725, 6.87581691905059560806756997204, 7.931702908710211417573937074151, 8.882289782398650893899368565683, 9.210986761182075897454567073949
