| L(s) = 1 | + 2.61·2-s + 3-s + 4.82·4-s + 0.712·5-s + 2.61·6-s − 2.05·7-s + 7.39·8-s + 9-s + 1.86·10-s − 4.17·11-s + 4.82·12-s + 4.74·13-s − 5.37·14-s + 0.712·15-s + 9.66·16-s − 17-s + 2.61·18-s + 5.95·19-s + 3.43·20-s − 2.05·21-s − 10.9·22-s + 6.82·23-s + 7.39·24-s − 4.49·25-s + 12.4·26-s + 27-s − 9.93·28-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 0.577·3-s + 2.41·4-s + 0.318·5-s + 1.06·6-s − 0.777·7-s + 2.61·8-s + 0.333·9-s + 0.588·10-s − 1.25·11-s + 1.39·12-s + 1.31·13-s − 1.43·14-s + 0.183·15-s + 2.41·16-s − 0.242·17-s + 0.615·18-s + 1.36·19-s + 0.769·20-s − 0.448·21-s − 2.32·22-s + 1.42·23-s + 1.50·24-s − 0.898·25-s + 2.43·26-s + 0.192·27-s − 1.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.007764372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.007764372\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 5 | \( 1 - 0.712T + 5T^{2} \) |
| 7 | \( 1 + 2.05T + 7T^{2} \) |
| 11 | \( 1 + 4.17T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 19 | \( 1 - 5.95T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 + 0.752T + 31T^{2} \) |
| 37 | \( 1 - 1.48T + 37T^{2} \) |
| 41 | \( 1 + 9.01T + 41T^{2} \) |
| 43 | \( 1 - 7.03T + 43T^{2} \) |
| 47 | \( 1 - 7.93T + 47T^{2} \) |
| 53 | \( 1 + 6.69T + 53T^{2} \) |
| 59 | \( 1 - 0.609T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 5.38T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 83 | \( 1 - 4.95T + 83T^{2} \) |
| 89 | \( 1 + 0.296T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.248407153213330378528901379737, −7.39983222657600038730659174759, −6.78592860790093405720858715367, −5.99446321947003800945621933554, −5.43376696192814345062126751958, −4.66811664624743250472591299825, −3.74359937388885383600307230789, −3.04163377797764745245714997731, −2.62526870002514734488204317215, −1.34932712021753713849838617383,
1.34932712021753713849838617383, 2.62526870002514734488204317215, 3.04163377797764745245714997731, 3.74359937388885383600307230789, 4.66811664624743250472591299825, 5.43376696192814345062126751958, 5.99446321947003800945621933554, 6.78592860790093405720858715367, 7.39983222657600038730659174759, 8.248407153213330378528901379737
