| L(s) = 1 | + 1.55·2-s − 3-s + 0.420·4-s + 3.26·5-s − 1.55·6-s − 2.45·8-s + 9-s + 5.07·10-s − 5.43·11-s − 0.420·12-s + 5.23·13-s − 3.26·15-s − 4.66·16-s + 0.709·17-s + 1.55·18-s − 2.66·19-s + 1.37·20-s − 8.45·22-s + 5.52·23-s + 2.45·24-s + 5.65·25-s + 8.15·26-s − 27-s + 4.20·29-s − 5.07·30-s − 2.85·31-s − 2.34·32-s + ⋯ |
| L(s) = 1 | + 1.10·2-s − 0.577·3-s + 0.210·4-s + 1.45·5-s − 0.635·6-s − 0.868·8-s + 0.333·9-s + 1.60·10-s − 1.63·11-s − 0.121·12-s + 1.45·13-s − 0.842·15-s − 1.16·16-s + 0.172·17-s + 0.366·18-s − 0.612·19-s + 0.307·20-s − 1.80·22-s + 1.15·23-s + 0.501·24-s + 1.13·25-s + 1.59·26-s − 0.192·27-s + 0.781·29-s − 0.927·30-s − 0.512·31-s − 0.414·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.273878648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.273878648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 5 | \( 1 - 3.26T + 5T^{2} \) |
| 11 | \( 1 + 5.43T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 - 0.709T + 17T^{2} \) |
| 19 | \( 1 + 2.66T + 19T^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 29 | \( 1 - 4.20T + 29T^{2} \) |
| 31 | \( 1 + 2.85T + 31T^{2} \) |
| 37 | \( 1 - 4.71T + 37T^{2} \) |
| 43 | \( 1 + 7.40T + 43T^{2} \) |
| 47 | \( 1 - 5.35T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 - 4.87T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 5.49T + 79T^{2} \) |
| 83 | \( 1 + 3.06T + 83T^{2} \) |
| 89 | \( 1 - 2.26T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.175646872187237510746522927063, −6.90398381047957707030618640249, −6.40577250707525691697971193353, −5.58105535123066230553099739073, −5.44204521071677116259402333685, −4.69650218442309712983687807722, −3.74147773544342773757530581789, −2.83284685196319799036230257306, −2.11391897553107234230142953865, −0.830317514254722618983451152838,
0.830317514254722618983451152838, 2.11391897553107234230142953865, 2.83284685196319799036230257306, 3.74147773544342773757530581789, 4.69650218442309712983687807722, 5.44204521071677116259402333685, 5.58105535123066230553099739073, 6.40577250707525691697971193353, 6.90398381047957707030618640249, 8.175646872187237510746522927063
