| L(s) = 1 | − 2.72·2-s + 5.44·4-s + 4.61·7-s − 9.39·8-s + 1.87·11-s + 5.48·13-s − 12.5·14-s + 14.7·16-s + 5.74·17-s − 5.40·19-s − 5.12·22-s + 0.137·23-s − 14.9·26-s + 25.1·28-s − 29-s − 8.18·31-s − 21.4·32-s − 15.6·34-s − 4.45·37-s + 14.7·38-s + 0.215·41-s + 7.17·43-s + 10.2·44-s − 0.373·46-s + 9.91·47-s + 14.3·49-s + 29.8·52-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 2.72·4-s + 1.74·7-s − 3.32·8-s + 0.566·11-s + 1.52·13-s − 3.36·14-s + 3.68·16-s + 1.39·17-s − 1.23·19-s − 1.09·22-s + 0.0285·23-s − 2.93·26-s + 4.74·28-s − 0.185·29-s − 1.46·31-s − 3.78·32-s − 2.68·34-s − 0.732·37-s + 2.39·38-s + 0.0336·41-s + 1.09·43-s + 1.54·44-s − 0.0551·46-s + 1.44·47-s + 2.04·49-s + 4.13·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.273414158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273414158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 - 1.87T + 11T^{2} \) |
| 13 | \( 1 - 5.48T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 19 | \( 1 + 5.40T + 19T^{2} \) |
| 23 | \( 1 - 0.137T + 23T^{2} \) |
| 31 | \( 1 + 8.18T + 31T^{2} \) |
| 37 | \( 1 + 4.45T + 37T^{2} \) |
| 41 | \( 1 - 0.215T + 41T^{2} \) |
| 43 | \( 1 - 7.17T + 43T^{2} \) |
| 47 | \( 1 - 9.91T + 47T^{2} \) |
| 53 | \( 1 - 1.14T + 53T^{2} \) |
| 59 | \( 1 - 0.244T + 59T^{2} \) |
| 61 | \( 1 - 8.75T + 61T^{2} \) |
| 67 | \( 1 + 6.53T + 67T^{2} \) |
| 71 | \( 1 - 0.277T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 1.66T + 79T^{2} \) |
| 83 | \( 1 + 9.58T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 - 8.04T + 97T^{2} \) |
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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
−8.200890227026331111397525711636, −7.50494684233084347930297076556, −7.06141290085779919138485889219, −5.94290102302672864148098665443, −5.64308094231660329900891467983, −4.25779775562344866421552661227, −3.39190401251495459236629757118, −2.12937738259497397240850411503, −1.54613899976345772624498960227, −0.862286974189832740068328951826,
0.862286974189832740068328951826, 1.54613899976345772624498960227, 2.12937738259497397240850411503, 3.39190401251495459236629757118, 4.25779775562344866421552661227, 5.64308094231660329900891467983, 5.94290102302672864148098665443, 7.06141290085779919138485889219, 7.50494684233084347930297076556, 8.200890227026331111397525711636
