| L(s) = 1 | + 1.64·2-s + 3-s + 0.703·4-s + 3.28·5-s + 1.64·6-s − 2.13·8-s + 9-s + 5.39·10-s − 3.35·11-s + 0.703·12-s + 5.75·13-s + 3.28·15-s − 4.91·16-s + 6.12·17-s + 1.64·18-s + 3.14·19-s + 2.30·20-s − 5.52·22-s − 9.17·23-s − 2.13·24-s + 5.77·25-s + 9.46·26-s + 27-s + 0.986·29-s + 5.39·30-s − 4.50·31-s − 3.81·32-s + ⋯ |
| L(s) = 1 | + 1.16·2-s + 0.577·3-s + 0.351·4-s + 1.46·5-s + 0.671·6-s − 0.753·8-s + 0.333·9-s + 1.70·10-s − 1.01·11-s + 0.203·12-s + 1.59·13-s + 0.847·15-s − 1.22·16-s + 1.48·17-s + 0.387·18-s + 0.720·19-s + 0.516·20-s − 1.17·22-s − 1.91·23-s − 0.435·24-s + 1.15·25-s + 1.85·26-s + 0.192·27-s + 0.183·29-s + 0.985·30-s − 0.808·31-s − 0.673·32-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
| \(L(1)\) |
\(\approx\) |
\(5.953693933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.953693933\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.64T + 2T^{2} \) |
| 5 | \( 1 - 3.28T + 5T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 - 5.75T + 13T^{2} \) |
| 17 | \( 1 - 6.12T + 17T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 + 9.17T + 23T^{2} \) |
| 29 | \( 1 - 0.986T + 29T^{2} \) |
| 31 | \( 1 + 4.50T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 43 | \( 1 - 7.67T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 2.61T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 1.80T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 5.92T + 71T^{2} \) |
| 73 | \( 1 + 9.66T + 73T^{2} \) |
| 79 | \( 1 + 8.44T + 79T^{2} \) |
| 83 | \( 1 - 5.92T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.025323616299624433510768745741, −7.36551620968759150021529444917, −6.06249860139614085810299852224, −5.89598842394088011258601660925, −5.43754807907829032527727575244, −4.35947814617686247353552591795, −3.66255108349273012079842136790, −2.87930595259475218377176235100, −2.19998535241385337635285298755, −1.12102967039295493655511340729,
1.12102967039295493655511340729, 2.19998535241385337635285298755, 2.87930595259475218377176235100, 3.66255108349273012079842136790, 4.35947814617686247353552591795, 5.43754807907829032527727575244, 5.89598842394088011258601660925, 6.06249860139614085810299852224, 7.36551620968759150021529444917, 8.025323616299624433510768745741
