| L(s) = 1 | + 4·7-s + 13-s + 3·19-s + 4·23-s − 29-s − 8·31-s + 37-s − 41-s − 6·43-s + 11·47-s + 9·49-s − 3·53-s + 10·59-s − 4·61-s − 13·67-s − 9·71-s − 3·79-s + 2·83-s − 10·89-s + 4·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.277·13-s + 0.688·19-s + 0.834·23-s − 0.185·29-s − 1.43·31-s + 0.164·37-s − 0.156·41-s − 0.914·43-s + 1.60·47-s + 9/7·49-s − 0.412·53-s + 1.30·59-s − 0.512·61-s − 1.58·67-s − 1.06·71-s − 0.337·79-s + 0.219·83-s − 1.05·89-s + 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.43146999270196, −12.98167602444192, −12.26822016867519, −11.93264992488980, −11.45229644953002, −10.98135366395947, −10.71773146917626, −10.16000608100499, −9.452834813310692, −9.041749855103274, −8.629532029013491, −8.029072837113908, −7.708043214285667, −7.064642404088268, −6.841143518658721, −5.816335988584705, −5.537041448865615, −5.107044115125091, −4.426547141221413, −4.094241339894232, −3.320832272127157, −2.790951768435154, −2.023547524747143, −1.485103844071565, −1.019966954634053, 0,
1.019966954634053, 1.485103844071565, 2.023547524747143, 2.790951768435154, 3.320832272127157, 4.094241339894232, 4.426547141221413, 5.107044115125091, 5.537041448865615, 5.816335988584705, 6.841143518658721, 7.064642404088268, 7.708043214285667, 8.029072837113908, 8.629532029013491, 9.041749855103274, 9.452834813310692, 10.16000608100499, 10.71773146917626, 10.98135366395947, 11.45229644953002, 11.93264992488980, 12.26822016867519, 12.98167602444192, 13.43146999270196
