| L(s) = 1 | + 9·3-s + 41.9·5-s − 134.·7-s + 81·9-s − 423.·11-s + 797.·13-s + 377.·15-s − 2.10e3·17-s − 1.56e3·19-s − 1.21e3·21-s + 529·23-s − 1.36e3·25-s + 729·27-s + 5.24e3·29-s + 1.85e3·31-s − 3.80e3·33-s − 5.65e3·35-s − 64.5·37-s + 7.17e3·39-s + 7.20e3·41-s + 4.27e3·43-s + 3.39e3·45-s + 1.60e4·47-s + 1.37e3·49-s − 1.89e4·51-s + 1.04e4·53-s − 1.77e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.750·5-s − 1.04·7-s + 0.333·9-s − 1.05·11-s + 1.30·13-s + 0.433·15-s − 1.76·17-s − 0.991·19-s − 0.600·21-s + 0.208·23-s − 0.437·25-s + 0.192·27-s + 1.15·29-s + 0.347·31-s − 0.608·33-s − 0.780·35-s − 0.00774·37-s + 0.755·39-s + 0.669·41-s + 0.352·43-s + 0.250·45-s + 1.06·47-s + 0.0821·49-s − 1.01·51-s + 0.510·53-s − 0.790·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.374637492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.374637492\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 23 | \( 1 - 529T \) |
| good | 5 | \( 1 - 41.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 134.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 423.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 797.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.56e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 5.24e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.85e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 64.5T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.20e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.27e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.60e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.04e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.29e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.44e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.35e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.72e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.01e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.73e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.57e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.28e5T + 8.58e9T^{2} \) |
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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
−8.891776053772851952255796539436, −8.657796878919653368748755150545, −7.46756825816313466350761851843, −6.38902199699190752487245908141, −6.07396795575661640586817326130, −4.72919882870839667204800729684, −3.77734117271254074043021453750, −2.69551134716554182697756913867, −2.06137342131504449830337996757, −0.61254075380633404277395220950,
0.61254075380633404277395220950, 2.06137342131504449830337996757, 2.69551134716554182697756913867, 3.77734117271254074043021453750, 4.72919882870839667204800729684, 6.07396795575661640586817326130, 6.38902199699190752487245908141, 7.46756825816313466350761851843, 8.657796878919653368748755150545, 8.891776053772851952255796539436
