| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s + 4·11-s + 12-s + 2·15-s + 16-s + 17-s − 18-s − 4·19-s + 2·20-s − 4·22-s − 24-s − 25-s + 27-s − 10·29-s − 2·30-s − 8·31-s − 32-s + 4·33-s − 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s − 0.852·22-s − 0.204·24-s − 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.365·30-s − 1.43·31-s − 0.176·32-s + 0.696·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.37093507540500, −15.49125810253492, −15.03148275375512, −14.43259440877750, −14.15823184760794, −13.33683035721845, −12.89634998719753, −12.29904811521619, −11.55834118216486, −11.00506666177533, −10.42587366837078, −9.749352835997431, −9.233801946998960, −9.031796235988731, −8.290837600143557, −7.504240649193244, −7.110529222742286, −6.210877895826569, −5.938509123935879, −5.041834408915526, −4.021232066278971, −3.568049750426744, −2.568439085766595, −1.806680463635172, −1.430099863010038, 0,
1.430099863010038, 1.806680463635172, 2.568439085766595, 3.568049750426744, 4.021232066278971, 5.041834408915526, 5.938509123935879, 6.210877895826569, 7.110529222742286, 7.504240649193244, 8.290837600143557, 9.031796235988731, 9.233801946998960, 9.749352835997431, 10.42587366837078, 11.00506666177533, 11.55834118216486, 12.29904811521619, 12.89634998719753, 13.33683035721845, 14.15823184760794, 14.43259440877750, 15.03148275375512, 15.49125810253492, 16.37093507540500
