| L(s) = 1 | − 3-s − 5-s − 4.88·7-s + 9-s − 6.11·11-s + 13-s + 15-s − 2.88·17-s + 7.00·19-s + 4.88·21-s − 4.88·23-s + 25-s − 27-s − 3.23·29-s − 9.77·31-s + 6.11·33-s + 4.88·35-s − 9.89·37-s − 39-s − 9.89·41-s − 4·43-s − 45-s + 5.77·47-s + 16.8·49-s + 2.88·51-s − 5.65·53-s + 6.11·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.84·7-s + 0.333·9-s − 1.84·11-s + 0.277·13-s + 0.258·15-s − 0.700·17-s + 1.60·19-s + 1.06·21-s − 1.01·23-s + 0.200·25-s − 0.192·27-s − 0.599·29-s − 1.75·31-s + 1.06·33-s + 0.826·35-s − 1.62·37-s − 0.160·39-s − 1.54·41-s − 0.609·43-s − 0.149·45-s + 0.842·47-s + 2.41·49-s + 0.404·51-s − 0.777·53-s + 0.825·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2919404005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2919404005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4.88T + 7T^{2} \) |
| 11 | \( 1 + 6.11T + 11T^{2} \) |
| 17 | \( 1 + 2.88T + 17T^{2} \) |
| 19 | \( 1 - 7.00T + 19T^{2} \) |
| 23 | \( 1 + 4.88T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 + 9.77T + 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 5.77T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 6.46T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 1.88T + 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 5.11T + 97T^{2} \) |
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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
−8.744826827990647677562113590328, −7.76050180586840933946459683473, −7.14983201578287301506150767879, −6.48544803677123905548403630156, −5.51582617237542203992457431357, −5.15222154151079267290819129672, −3.70431914122531738924342712761, −3.30465206974145275501544298790, −2.14713130507221737127643673172, −0.31174588601431505970251412859,
0.31174588601431505970251412859, 2.14713130507221737127643673172, 3.30465206974145275501544298790, 3.70431914122531738924342712761, 5.15222154151079267290819129672, 5.51582617237542203992457431357, 6.48544803677123905548403630156, 7.14983201578287301506150767879, 7.76050180586840933946459683473, 8.744826827990647677562113590328
