| L(s) = 1 | − 2.34·3-s − 0.153·5-s + 2.78·7-s + 2.48·9-s + 0.333·11-s − 4.04·13-s + 0.359·15-s + 2.90·17-s + 3.30·19-s − 6.51·21-s + 1.15·23-s − 4.97·25-s + 1.21·27-s + 1.22·29-s + 5.46·31-s − 0.780·33-s − 0.426·35-s − 0.439·37-s + 9.46·39-s − 4.24·41-s − 43-s − 0.380·45-s + 1.58·47-s + 0.733·49-s − 6.80·51-s + 3.18·53-s − 0.0512·55-s + ⋯ |
| L(s) = 1 | − 1.35·3-s − 0.0686·5-s + 1.05·7-s + 0.827·9-s + 0.100·11-s − 1.12·13-s + 0.0928·15-s + 0.705·17-s + 0.759·19-s − 1.42·21-s + 0.241·23-s − 0.995·25-s + 0.233·27-s + 0.226·29-s + 0.981·31-s − 0.135·33-s − 0.0721·35-s − 0.0722·37-s + 1.51·39-s − 0.662·41-s − 0.152·43-s − 0.0567·45-s + 0.231·47-s + 0.104·49-s − 0.953·51-s + 0.437·53-s − 0.00690·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.107309471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107309471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 2.34T + 3T^{2} \) |
| 5 | \( 1 + 0.153T + 5T^{2} \) |
| 7 | \( 1 - 2.78T + 7T^{2} \) |
| 11 | \( 1 - 0.333T + 11T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 - 2.90T + 17T^{2} \) |
| 19 | \( 1 - 3.30T + 19T^{2} \) |
| 23 | \( 1 - 1.15T + 23T^{2} \) |
| 29 | \( 1 - 1.22T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 0.439T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 - 3.18T + 53T^{2} \) |
| 59 | \( 1 + 5.81T + 59T^{2} \) |
| 61 | \( 1 + 4.48T + 61T^{2} \) |
| 67 | \( 1 + 1.19T + 67T^{2} \) |
| 71 | \( 1 + 1.93T + 71T^{2} \) |
| 73 | \( 1 + 1.64T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 0.520T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 6.07T + 97T^{2} \) |
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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
−8.782870093022940680604760866840, −7.83711009081753941559296860164, −7.35418582385832637394602819326, −6.40106941148000513239313292126, −5.62136599363664554610637558170, −4.98419867869397980779655987433, −4.46827687709930224933392448552, −3.15036660280041724969022807138, −1.84792558380617208545166714983, −0.71449525337391997778079086562,
0.71449525337391997778079086562, 1.84792558380617208545166714983, 3.15036660280041724969022807138, 4.46827687709930224933392448552, 4.98419867869397980779655987433, 5.62136599363664554610637558170, 6.40106941148000513239313292126, 7.35418582385832637394602819326, 7.83711009081753941559296860164, 8.782870093022940680604760866840
