| L(s) = 1 | + 2-s − 3-s + 4-s − 4.16·5-s − 6-s + 0.910·7-s + 8-s + 9-s − 4.16·10-s − 1.61·11-s − 12-s + 5.05·13-s + 0.910·14-s + 4.16·15-s + 16-s − 5.02·17-s + 18-s − 19-s − 4.16·20-s − 0.910·21-s − 1.61·22-s − 6.18·23-s − 24-s + 12.3·25-s + 5.05·26-s − 27-s + 0.910·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.86·5-s − 0.408·6-s + 0.344·7-s + 0.353·8-s + 0.333·9-s − 1.31·10-s − 0.486·11-s − 0.288·12-s + 1.40·13-s + 0.243·14-s + 1.07·15-s + 0.250·16-s − 1.21·17-s + 0.235·18-s − 0.229·19-s − 0.930·20-s − 0.198·21-s − 0.344·22-s − 1.28·23-s − 0.204·24-s + 2.46·25-s + 0.991·26-s − 0.192·27-s + 0.172·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 + 4.16T + 5T^{2} \) |
| 7 | \( 1 - 0.910T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 - 5.05T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 23 | \( 1 + 6.18T + 23T^{2} \) |
| 29 | \( 1 - 6.09T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 1.16T + 37T^{2} \) |
| 41 | \( 1 + 3.25T + 41T^{2} \) |
| 43 | \( 1 + 9.91T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 59 | \( 1 + 0.621T + 59T^{2} \) |
| 61 | \( 1 - 6.45T + 61T^{2} \) |
| 67 | \( 1 - 8.64T + 67T^{2} \) |
| 71 | \( 1 + 6.55T + 71T^{2} \) |
| 73 | \( 1 + 2.69T + 73T^{2} \) |
| 79 | \( 1 - 8.95T + 79T^{2} \) |
| 83 | \( 1 + 4.45T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.78624392998773214076017990831, −6.68675130841443971750238348071, −6.54117651096292174654147259107, −5.43661473564699417888968296796, −4.57076441235566882751798504885, −4.21454909354591238494941596514, −3.51949989738778532234850728535, −2.55205431550947851623001798224, −1.18802453202495941393689572359, 0,
1.18802453202495941393689572359, 2.55205431550947851623001798224, 3.51949989738778532234850728535, 4.21454909354591238494941596514, 4.57076441235566882751798504885, 5.43661473564699417888968296796, 6.54117651096292174654147259107, 6.68675130841443971750238348071, 7.78624392998773214076017990831
