| L(s) = 1 | − 2.59·3-s − 1.59·5-s + 4.34·7-s + 3.74·9-s − 1.74·11-s − 0.748·13-s + 4.15·15-s + 3·17-s + 19-s − 11.2·21-s − 1.94·23-s − 2.44·25-s − 1.94·27-s + 7.28·29-s + 0.654·31-s + 4.54·33-s − 6.94·35-s − 3.84·37-s + 1.94·39-s + 4.54·41-s + 10.4·43-s − 5.98·45-s + 7.59·47-s + 11.8·49-s − 7.79·51-s − 6.59·53-s + 2.79·55-s + ⋯ |
| L(s) = 1 | − 1.49·3-s − 0.714·5-s + 1.64·7-s + 1.24·9-s − 0.527·11-s − 0.207·13-s + 1.07·15-s + 0.727·17-s + 0.229·19-s − 2.46·21-s − 0.405·23-s − 0.489·25-s − 0.374·27-s + 1.35·29-s + 0.117·31-s + 0.790·33-s − 1.17·35-s − 0.632·37-s + 0.311·39-s + 0.709·41-s + 1.59·43-s − 0.892·45-s + 1.10·47-s + 1.69·49-s − 1.09·51-s − 0.906·53-s + 0.376·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.047264829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.047264829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.59T + 3T^{2} \) |
| 5 | \( 1 + 1.59T + 5T^{2} \) |
| 7 | \( 1 - 4.34T + 7T^{2} \) |
| 11 | \( 1 + 1.74T + 11T^{2} \) |
| 13 | \( 1 + 0.748T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 23 | \( 1 + 1.94T + 23T^{2} \) |
| 29 | \( 1 - 7.28T + 29T^{2} \) |
| 31 | \( 1 - 0.654T + 31T^{2} \) |
| 37 | \( 1 + 3.84T + 37T^{2} \) |
| 41 | \( 1 - 4.54T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 7.59T + 47T^{2} \) |
| 53 | \( 1 + 6.59T + 53T^{2} \) |
| 59 | \( 1 - 6.74T + 59T^{2} \) |
| 61 | \( 1 + 9.09T + 61T^{2} \) |
| 67 | \( 1 + 7.40T + 67T^{2} \) |
| 71 | \( 1 + 1.69T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 - 0.654T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.960178624289310425314481785782, −7.65766857832927073742057659782, −6.86370768679898231799389097012, −5.76001690951863687085290660334, −5.48695209062501459826212671126, −4.53391655966380494956904589414, −4.26659547311492724948304595680, −2.83688522086771593417871924833, −1.57777762221849269514229729347, −0.64400398787005034526729811451,
0.64400398787005034526729811451, 1.57777762221849269514229729347, 2.83688522086771593417871924833, 4.26659547311492724948304595680, 4.53391655966380494956904589414, 5.48695209062501459826212671126, 5.76001690951863687085290660334, 6.86370768679898231799389097012, 7.65766857832927073742057659782, 7.960178624289310425314481785782
