L(s) = 1 | + 3-s − 2·5-s + 9-s + 11-s − 2·15-s + 6·17-s − 4·23-s − 25-s + 27-s + 2·29-s + 33-s + 10·37-s − 6·41-s + 8·43-s − 2·45-s − 4·47-s − 7·49-s + 6·51-s − 6·53-s − 2·55-s − 12·59-s + 2·61-s + 4·67-s − 4·69-s + 12·71-s + 14·73-s − 75-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s − 0.516·15-s + 1.45·17-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.174·33-s + 1.64·37-s − 0.937·41-s + 1.21·43-s − 0.298·45-s − 0.583·47-s − 49-s + 0.840·51-s − 0.824·53-s − 0.269·55-s − 1.56·59-s + 0.256·61-s + 0.488·67-s − 0.481·69-s + 1.42·71-s + 1.63·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.17770892532491, −13.86090733393766, −12.98691006991660, −12.67728208168969, −12.15985331309007, −11.68633358439305, −11.27236825611316, −10.66985312617094, −10.03374512483535, −9.555676769459763, −9.298635344656335, −8.321600122193578, −8.006235060902457, −7.857362914920932, −7.097696620870330, −6.571341119510701, −5.903189133260121, −5.401404514985841, −4.506114865716731, −4.242320148319004, −3.476615286872492, −3.178747324979824, −2.411480376634791, −1.601838698305269, −0.9449004165021365, 0,
0.9449004165021365, 1.601838698305269, 2.411480376634791, 3.178747324979824, 3.476615286872492, 4.242320148319004, 4.506114865716731, 5.401404514985841, 5.903189133260121, 6.571341119510701, 7.097696620870330, 7.857362914920932, 8.006235060902457, 8.321600122193578, 9.298635344656335, 9.555676769459763, 10.03374512483535, 10.66985312617094, 11.27236825611316, 11.68633358439305, 12.15985331309007, 12.67728208168969, 12.98691006991660, 13.86090733393766, 14.17770892532491