| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 2·11-s − 12-s − 4·13-s − 14-s + 16-s + 6·17-s + 18-s − 4·19-s + 21-s − 2·22-s + 23-s − 24-s − 4·26-s − 27-s − 28-s + 2·29-s + 2·31-s + 32-s + 2·33-s + 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.218·21-s − 0.426·22-s + 0.208·23-s − 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.359·31-s + 0.176·32-s + 0.348·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.65902256737308, −14.97065545137851, −14.79028989352531, −14.01723243577831, −13.47550119856448, −12.94912627906690, −12.25316286943022, −12.19038053015667, −11.53138556101004, −10.71978373973217, −10.26000219049835, −9.970094027494530, −9.158209073940926, −8.338843769034456, −7.741796670153187, −7.127060897529182, −6.657729597989804, −5.916613793042345, −5.362002670639294, −4.937020488382258, −4.202062135125011, −3.483204423093327, −2.755589921450941, −2.109100373756922, −1.042560095208731, 0,
1.042560095208731, 2.109100373756922, 2.755589921450941, 3.483204423093327, 4.202062135125011, 4.937020488382258, 5.362002670639294, 5.916613793042345, 6.657729597989804, 7.127060897529182, 7.741796670153187, 8.338843769034456, 9.158209073940926, 9.970094027494530, 10.26000219049835, 10.71978373973217, 11.53138556101004, 12.19038053015667, 12.25316286943022, 12.94912627906690, 13.47550119856448, 14.01723243577831, 14.79028989352531, 14.97065545137851, 15.65902256737308
