| L(s) = 1 | + 3-s − 1.66·5-s − 0.368·7-s + 9-s + 2.19·11-s + 0.456·13-s − 1.66·15-s − 4.55·17-s + 1.18·19-s − 0.368·21-s + 4.16·23-s − 2.23·25-s + 27-s − 4.37·29-s − 3.77·31-s + 2.19·33-s + 0.612·35-s + 7.10·37-s + 0.456·39-s + 3.74·41-s − 0.00494·43-s − 1.66·45-s + 12.8·47-s − 6.86·49-s − 4.55·51-s − 1.58·53-s − 3.64·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.743·5-s − 0.139·7-s + 0.333·9-s + 0.661·11-s + 0.126·13-s − 0.429·15-s − 1.10·17-s + 0.271·19-s − 0.0803·21-s + 0.868·23-s − 0.446·25-s + 0.192·27-s − 0.811·29-s − 0.678·31-s + 0.381·33-s + 0.103·35-s + 1.16·37-s + 0.0730·39-s + 0.584·41-s − 0.000754·43-s − 0.247·45-s + 1.87·47-s − 0.980·49-s − 0.638·51-s − 0.217·53-s − 0.491·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.014950552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.014950552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 + 0.368T + 7T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 13 | \( 1 - 0.456T + 13T^{2} \) |
| 17 | \( 1 + 4.55T + 17T^{2} \) |
| 19 | \( 1 - 1.18T + 19T^{2} \) |
| 23 | \( 1 - 4.16T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 + 3.77T + 31T^{2} \) |
| 37 | \( 1 - 7.10T + 37T^{2} \) |
| 41 | \( 1 - 3.74T + 41T^{2} \) |
| 43 | \( 1 + 0.00494T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + 1.58T + 53T^{2} \) |
| 59 | \( 1 - 9.98T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 9.92T + 67T^{2} \) |
| 71 | \( 1 + 7.66T + 71T^{2} \) |
| 73 | \( 1 + 7.07T + 73T^{2} \) |
| 79 | \( 1 + 2.49T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 4.80T + 89T^{2} \) |
| 97 | \( 1 + 7.92T + 97T^{2} \) |
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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
−8.034210116912222295948335973679, −7.40769683844926531533787705758, −6.84362387615766915016247763557, −6.03547533380724200589621558766, −5.11131826754923543742421105187, −4.11948284001224947877498963712, −3.82431312015230834777264755026, −2.82212159018363312506827096885, −1.94672314369756365679321517359, −0.72254269750980156689782402128,
0.72254269750980156689782402128, 1.94672314369756365679321517359, 2.82212159018363312506827096885, 3.82431312015230834777264755026, 4.11948284001224947877498963712, 5.11131826754923543742421105187, 6.03547533380724200589621558766, 6.84362387615766915016247763557, 7.40769683844926531533787705758, 8.034210116912222295948335973679
