| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 2·7-s − 8-s + 9-s + 12-s + 2·13-s + 2·14-s + 16-s + 6·17-s − 18-s − 2·21-s − 6·23-s − 24-s − 2·26-s + 27-s − 2·28-s + 4·29-s − 32-s − 6·34-s + 36-s + 10·37-s + 2·39-s + 8·41-s + 2·42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.436·21-s − 1.25·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.377·28-s + 0.742·29-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 1.64·37-s + 0.320·39-s + 1.24·41-s + 0.308·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.198111827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.198111827\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.44987595689119, −14.10957185074011, −13.21497317664114, −13.07386926239970, −12.27307519076226, −11.96586304477372, −11.30059265806240, −10.68948563167616, −10.07687008136090, −9.769594058058828, −9.380880940385910, −8.651797026249202, −8.183959601426114, −7.738934460258048, −7.243364457880370, −6.473445547739808, −6.047978967560342, −5.560159167554416, −4.582159173304203, −3.909697290357201, −3.348600448222001, −2.774235491531735, −2.121503023408737, −1.235344989979892, −0.6083823631895613,
0.6083823631895613, 1.235344989979892, 2.121503023408737, 2.774235491531735, 3.348600448222001, 3.909697290357201, 4.582159173304203, 5.560159167554416, 6.047978967560342, 6.473445547739808, 7.243364457880370, 7.738934460258048, 8.183959601426114, 8.651797026249202, 9.380880940385910, 9.769594058058828, 10.07687008136090, 10.68948563167616, 11.30059265806240, 11.96586304477372, 12.27307519076226, 13.07386926239970, 13.21497317664114, 14.10957185074011, 14.44987595689119
