| L(s) = 1 | − 1.41·2-s + 2.64·5-s − 7-s + 2.82·8-s − 3.74·10-s − 1.74·13-s + 1.41·14-s − 4.00·16-s − 0.182·17-s + 1.74·19-s − 6.70·23-s + 2.00·25-s + 2.46·26-s + 6.70·29-s − 9.48·31-s + 0.258·34-s − 2.64·35-s − 11.4·37-s − 2.46·38-s + 7.48·40-s + 5.65·41-s + 43-s + 9.48·46-s + 0.182·47-s + 49-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 1.18·5-s − 0.377·7-s + 0.999·8-s − 1.18·10-s − 0.483·13-s + 0.377·14-s − 1.00·16-s − 0.0443·17-s + 0.399·19-s − 1.39·23-s + 0.400·25-s + 0.483·26-s + 1.24·29-s − 1.70·31-s + 0.0443·34-s − 0.447·35-s − 1.88·37-s − 0.399·38-s + 1.18·40-s + 0.883·41-s + 0.152·43-s + 1.39·46-s + 0.0266·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.021865349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.021865349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 5 | \( 1 - 2.64T + 5T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 + 0.182T + 17T^{2} \) |
| 19 | \( 1 - 1.74T + 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 9.48T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 0.182T + 47T^{2} \) |
| 53 | \( 1 - 7.75T + 53T^{2} \) |
| 59 | \( 1 + 3.01T + 59T^{2} \) |
| 61 | \( 1 - 5.74T + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 - 1.41T + 71T^{2} \) |
| 73 | \( 1 - 8.25T + 73T^{2} \) |
| 79 | \( 1 + 1.48T + 79T^{2} \) |
| 83 | \( 1 + 5.83T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.997530156527026843393214455415, −7.24323471603747949333871657986, −6.67178680657557094313847605107, −5.75284848544932607658528513583, −5.27433428975415827722478909487, −4.34264998265708567756820058587, −3.46370126450760537986432899182, −2.27081008589983561823246724631, −1.76219906738499970597008545326, −0.58881439203640554331968930877,
0.58881439203640554331968930877, 1.76219906738499970597008545326, 2.27081008589983561823246724631, 3.46370126450760537986432899182, 4.34264998265708567756820058587, 5.27433428975415827722478909487, 5.75284848544932607658528513583, 6.67178680657557094313847605107, 7.24323471603747949333871657986, 7.997530156527026843393214455415
