L(s) = 1 | − 5-s + 4·7-s + 2·13-s + 2·17-s − 4·19-s − 4·23-s + 25-s − 2·29-s − 8·31-s − 4·35-s + 6·37-s − 6·41-s + 8·43-s − 4·47-s + 9·49-s − 6·53-s + 4·59-s + 2·61-s − 2·65-s + 8·67-s + 6·73-s − 16·83-s − 2·85-s + 6·89-s + 8·91-s + 4·95-s − 14·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.51·7-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.676·35-s + 0.986·37-s − 0.937·41-s + 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.520·59-s + 0.256·61-s − 0.248·65-s + 0.977·67-s + 0.702·73-s − 1.75·83-s − 0.216·85-s + 0.635·89-s + 0.838·91-s + 0.410·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.83024770922495, −14.47673237636014, −14.10259529571048, −13.40850051127192, −12.78538835313349, −12.39764274170998, −11.67730353800334, −11.28813755736317, −10.92268349597340, −10.40847266019430, −9.662637701150148, −9.081676908409405, −8.385526582652375, −8.105891802241959, −7.639302302410770, −6.989193532005891, −6.302968568930480, −5.587900928321149, −5.191496443141863, −4.352713102987741, −4.043040138408595, −3.333766459040656, −2.350831722555156, −1.773874121610584, −1.088448086118744, 0,
1.088448086118744, 1.773874121610584, 2.350831722555156, 3.333766459040656, 4.043040138408595, 4.352713102987741, 5.191496443141863, 5.587900928321149, 6.302968568930480, 6.989193532005891, 7.639302302410770, 8.105891802241959, 8.385526582652375, 9.081676908409405, 9.662637701150148, 10.40847266019430, 10.92268349597340, 11.28813755736317, 11.67730353800334, 12.39764274170998, 12.78538835313349, 13.40850051127192, 14.10259529571048, 14.47673237636014, 14.83024770922495