| L(s) = 1 | + 2-s + 0.445·3-s + 4-s − 0.116·5-s + 0.445·6-s − 4.25·7-s + 8-s − 2.80·9-s − 0.116·10-s + 4.46·11-s + 0.445·12-s + 1.42·13-s − 4.25·14-s − 0.0516·15-s + 16-s − 6.60·17-s − 2.80·18-s + 6.67·19-s − 0.116·20-s − 1.89·21-s + 4.46·22-s + 23-s + 0.445·24-s − 4.98·25-s + 1.42·26-s − 2.58·27-s − 4.25·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.257·3-s + 0.5·4-s − 0.0519·5-s + 0.181·6-s − 1.60·7-s + 0.353·8-s − 0.933·9-s − 0.0367·10-s + 1.34·11-s + 0.128·12-s + 0.396·13-s − 1.13·14-s − 0.0133·15-s + 0.250·16-s − 1.60·17-s − 0.660·18-s + 1.53·19-s − 0.0259·20-s − 0.413·21-s + 0.950·22-s + 0.208·23-s + 0.0908·24-s − 0.997·25-s + 0.280·26-s − 0.497·27-s − 0.804·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.654144284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.654144284\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 0.445T + 3T^{2} \) |
| 5 | \( 1 + 0.116T + 5T^{2} \) |
| 7 | \( 1 + 4.25T + 7T^{2} \) |
| 11 | \( 1 - 4.46T + 11T^{2} \) |
| 13 | \( 1 - 1.42T + 13T^{2} \) |
| 17 | \( 1 + 6.60T + 17T^{2} \) |
| 19 | \( 1 - 6.67T + 19T^{2} \) |
| 29 | \( 1 - 4.04T + 29T^{2} \) |
| 31 | \( 1 - 5.81T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 2.99T + 53T^{2} \) |
| 59 | \( 1 + 5.45T + 59T^{2} \) |
| 61 | \( 1 - 8.23T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 6.59T + 71T^{2} \) |
| 73 | \( 1 + 2.03T + 73T^{2} \) |
| 79 | \( 1 - 0.536T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 3.51T + 89T^{2} \) |
| 97 | \( 1 + 0.967T + 97T^{2} \) |
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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
−8.067542034692227280250665447139, −7.02488405465733440094004420855, −6.57738485993939008229706169173, −6.05645962609867075067355921068, −5.30087171339928994107019661580, −4.20675404504066279324909619787, −3.59684865430132083287674523848, −3.01513792562118300304057605285, −2.16698182170321475890457359469, −0.73859772462031495157873142389,
0.73859772462031495157873142389, 2.16698182170321475890457359469, 3.01513792562118300304057605285, 3.59684865430132083287674523848, 4.20675404504066279324909619787, 5.30087171339928994107019661580, 6.05645962609867075067355921068, 6.57738485993939008229706169173, 7.02488405465733440094004420855, 8.067542034692227280250665447139
