| L(s) = 1 | + 2·5-s − 6·13-s − 17-s + 2·19-s − 25-s − 8·29-s + 2·37-s + 2·41-s + 8·43-s − 8·47-s + 2·53-s + 12·59-s + 4·61-s − 12·65-s + 12·67-s − 8·73-s − 8·79-s − 2·85-s + 10·89-s + 4·95-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.66·13-s − 0.242·17-s + 0.458·19-s − 1/5·25-s − 1.48·29-s + 0.328·37-s + 0.312·41-s + 1.21·43-s − 1.16·47-s + 0.274·53-s + 1.56·59-s + 0.512·61-s − 1.48·65-s + 1.46·67-s − 0.936·73-s − 0.900·79-s − 0.216·85-s + 1.05·89-s + 0.410·95-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.944886699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944886699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.92033386886470, −14.60747275354898, −14.19019192091677, −13.49420818976538, −13.02560512655082, −12.61903082830999, −11.92548308701613, −11.42695364775616, −10.86693941565370, −10.07181652922708, −9.740348274542234, −9.387423472243391, −8.713832390062314, −7.922595582493134, −7.399627663538299, −6.915647770174595, −6.194942068017204, −5.474567590415255, −5.231014158596817, −4.381544210874853, −3.737985486332155, −2.767691867071923, −2.281794252799006, −1.633748226340058, −0.5145497515938038,
0.5145497515938038, 1.633748226340058, 2.281794252799006, 2.767691867071923, 3.737985486332155, 4.381544210874853, 5.231014158596817, 5.474567590415255, 6.194942068017204, 6.915647770174595, 7.399627663538299, 7.922595582493134, 8.713832390062314, 9.387423472243391, 9.740348274542234, 10.07181652922708, 10.86693941565370, 11.42695364775616, 11.92548308701613, 12.61903082830999, 13.02560512655082, 13.49420818976538, 14.19019192091677, 14.60747275354898, 14.92033386886470
