| L(s) = 1 | + 2.47·2-s − 3-s + 4.11·4-s + 4.25·5-s − 2.47·6-s + 5.22·8-s + 9-s + 10.5·10-s − 0.0562·11-s − 4.11·12-s + 0.121·13-s − 4.25·15-s + 4.69·16-s + 6.50·17-s + 2.47·18-s − 2.99·19-s + 17.5·20-s − 0.139·22-s − 7.44·23-s − 5.22·24-s + 13.1·25-s + 0.300·26-s − 27-s + 6.83·29-s − 10.5·30-s + 5.89·31-s + 1.15·32-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 0.577·3-s + 2.05·4-s + 1.90·5-s − 1.00·6-s + 1.84·8-s + 0.333·9-s + 3.32·10-s − 0.0169·11-s − 1.18·12-s + 0.0336·13-s − 1.09·15-s + 1.17·16-s + 1.57·17-s + 0.582·18-s − 0.687·19-s + 3.91·20-s − 0.0296·22-s − 1.55·23-s − 1.06·24-s + 2.62·25-s + 0.0588·26-s − 0.192·27-s + 1.26·29-s − 1.92·30-s + 1.05·31-s + 0.204·32-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
| \(L(1)\) |
\(\approx\) |
\(7.613722055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.613722055\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.47T + 2T^{2} \) |
| 5 | \( 1 - 4.25T + 5T^{2} \) |
| 11 | \( 1 + 0.0562T + 11T^{2} \) |
| 13 | \( 1 - 0.121T + 13T^{2} \) |
| 17 | \( 1 - 6.50T + 17T^{2} \) |
| 19 | \( 1 + 2.99T + 19T^{2} \) |
| 23 | \( 1 + 7.44T + 23T^{2} \) |
| 29 | \( 1 - 6.83T + 29T^{2} \) |
| 31 | \( 1 - 5.89T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 2.94T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 9.07T + 61T^{2} \) |
| 67 | \( 1 - 2.83T + 67T^{2} \) |
| 71 | \( 1 + 2.41T + 71T^{2} \) |
| 73 | \( 1 + 7.14T + 73T^{2} \) |
| 79 | \( 1 - 8.01T + 79T^{2} \) |
| 83 | \( 1 - 3.64T + 83T^{2} \) |
| 89 | \( 1 - 7.76T + 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
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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
−7.81177867721507123030822441720, −6.76759446745398083428185537215, −6.27679431937099300679304084459, −5.89625129775021773408900846381, −5.21936426110948999197092401164, −4.74879036312062929526753961121, −3.75990908180229212404655736516, −2.84233029812256274906965739981, −2.11289687543937546916992617755, −1.28478882793688667311188016921,
1.28478882793688667311188016921, 2.11289687543937546916992617755, 2.84233029812256274906965739981, 3.75990908180229212404655736516, 4.74879036312062929526753961121, 5.21936426110948999197092401164, 5.89625129775021773408900846381, 6.27679431937099300679304084459, 6.76759446745398083428185537215, 7.81177867721507123030822441720
