| L(s) = 1 | − 5-s − 4·13-s − 4·19-s + 8·23-s + 25-s − 2·29-s + 4·31-s − 2·37-s − 10·41-s + 43-s − 12·47-s − 7·49-s − 2·53-s + 8·59-s + 12·61-s + 4·65-s − 8·67-s + 12·71-s − 6·73-s + 6·83-s − 12·89-s + 4·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.10·13-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s − 0.328·37-s − 1.56·41-s + 0.152·43-s − 1.75·47-s − 49-s − 0.274·53-s + 1.04·59-s + 1.53·61-s + 0.496·65-s − 0.977·67-s + 1.42·71-s − 0.702·73-s + 0.658·83-s − 1.27·89-s + 0.410·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.161484595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.161484595\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.94413094260769, −14.79454572931577, −14.17064785049116, −13.33270826041692, −13.01899912985441, −12.47084521727495, −11.90557547045261, −11.34934892909881, −10.94758572080646, −10.17661056406368, −9.802691870072741, −9.151523393728315, −8.428188087561120, −8.164871798375093, −7.316781548790886, −6.839485902131922, −6.449432612878456, −5.418652353057397, −4.947201588751332, −4.471873374715023, −3.600132851280405, −3.027403590520520, −2.294581764982286, −1.472452439758246, −0.4104509558882428,
0.4104509558882428, 1.472452439758246, 2.294581764982286, 3.027403590520520, 3.600132851280405, 4.471873374715023, 4.947201588751332, 5.418652353057397, 6.449432612878456, 6.839485902131922, 7.316781548790886, 8.164871798375093, 8.428188087561120, 9.151523393728315, 9.802691870072741, 10.17661056406368, 10.94758572080646, 11.34934892909881, 11.90557547045261, 12.47084521727495, 13.01899912985441, 13.33270826041692, 14.17064785049116, 14.79454572931577, 14.94413094260769
