| L(s) = 1 | + 2-s + 3-s + 4-s − 2.94·5-s + 6-s + 2.59·7-s + 8-s + 9-s − 2.94·10-s − 3.05·11-s + 12-s − 4.13·13-s + 2.59·14-s − 2.94·15-s + 16-s − 17-s + 18-s − 2.51·19-s − 2.94·20-s + 2.59·21-s − 3.05·22-s + 4.05·23-s + 24-s + 3.68·25-s − 4.13·26-s + 27-s + 2.59·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.31·5-s + 0.408·6-s + 0.980·7-s + 0.353·8-s + 0.333·9-s − 0.931·10-s − 0.922·11-s + 0.288·12-s − 1.14·13-s + 0.692·14-s − 0.760·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.575·19-s − 0.658·20-s + 0.565·21-s − 0.652·22-s + 0.845·23-s + 0.204·24-s + 0.736·25-s − 0.811·26-s + 0.192·27-s + 0.490·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 + 2.94T + 5T^{2} \) |
| 7 | \( 1 - 2.59T + 7T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 + 4.13T + 13T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 23 | \( 1 - 4.05T + 23T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 - 8.42T + 31T^{2} \) |
| 37 | \( 1 + 4.29T + 37T^{2} \) |
| 41 | \( 1 + 0.997T + 41T^{2} \) |
| 43 | \( 1 + 1.87T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 7.72T + 53T^{2} \) |
| 61 | \( 1 - 0.942T + 61T^{2} \) |
| 67 | \( 1 + 4.57T + 67T^{2} \) |
| 71 | \( 1 + 4.76T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 4.73T + 79T^{2} \) |
| 83 | \( 1 + 5.41T + 83T^{2} \) |
| 89 | \( 1 - 4.91T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−7.79885875084511952109330827852, −7.16860316260396005936953915772, −6.43538317167000143952206237394, −5.14107297684718682181283624500, −4.75214743815587515412968613661, −4.21521302604426858044841083053, −3.16787938207061625887660750624, −2.63521841315043912642928360461, −1.56441127454342634526107652731, 0,
1.56441127454342634526107652731, 2.63521841315043912642928360461, 3.16787938207061625887660750624, 4.21521302604426858044841083053, 4.75214743815587515412968613661, 5.14107297684718682181283624500, 6.43538317167000143952206237394, 7.16860316260396005936953915772, 7.79885875084511952109330827852
