| L(s) = 1 | − 5-s − 11-s + 5·13-s + 5·19-s + 6·23-s + 25-s − 31-s − 7·37-s − 43-s − 12·47-s + 6·53-s + 55-s − 6·59-s − 10·61-s − 5·65-s − 13·67-s − 12·71-s + 5·73-s − 79-s − 5·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 6·115-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 1.38·13-s + 1.14·19-s + 1.25·23-s + 1/5·25-s − 0.179·31-s − 1.15·37-s − 0.152·43-s − 1.75·47-s + 0.824·53-s + 0.134·55-s − 0.781·59-s − 1.28·61-s − 0.620·65-s − 1.58·67-s − 1.42·71-s + 0.585·73-s − 0.112·79-s − 0.512·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.559·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.154195293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.154195293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.77993880841411, −13.21510095248225, −13.00209712718312, −12.18983959066586, −11.87026067922829, −11.24134861249639, −10.96629613463605, −10.38179965267883, −9.914012473729736, −9.121124666194077, −8.863079610410703, −8.349144618407812, −7.685420602842065, −7.321974502950329, −6.736671008495697, −6.123615258400583, −5.630785804223135, −4.956314704208659, −4.570872373826679, −3.729154308773699, −3.224292719240764, −2.936977611690190, −1.775190146058172, −1.303838993376178, −0.4894409854683823,
0.4894409854683823, 1.303838993376178, 1.775190146058172, 2.936977611690190, 3.224292719240764, 3.729154308773699, 4.570872373826679, 4.956314704208659, 5.630785804223135, 6.123615258400583, 6.736671008495697, 7.321974502950329, 7.685420602842065, 8.349144618407812, 8.863079610410703, 9.121124666194077, 9.914012473729736, 10.38179965267883, 10.96629613463605, 11.24134861249639, 11.87026067922829, 12.18983959066586, 13.00209712718312, 13.21510095248225, 13.77993880841411
