L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 12-s + 14-s + 16-s + 6·17-s + 18-s + 4·19-s − 21-s − 24-s − 27-s + 28-s − 6·29-s + 4·31-s + 32-s + 6·34-s + 36-s + 2·37-s + 4·38-s − 6·41-s − 42-s − 8·43-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 − 0.288·12-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 − 0.204·24-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.648·38-s − 0.937·41-s − 0.154·42-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.23957982434998, −13.02956029114208, −12.46907894158106, −11.85943439844901, −11.60447466148275, −11.36710458175457, −10.57932305246922, −10.23394653129571, −9.603504233922508, −9.454823099621902, −8.309172276617504, −8.154051896209749, −7.571779540526922, −7.005129171962144, −6.568113524406891, −6.016840842917124, −5.352844880568423, −5.174732620979290, −4.718430329474392, −3.815113280296829, −3.552322552630517, −2.940398376294641, −2.166904282348720, −1.460329120747594, −1.016886220946540, 0,
1.016886220946540, 1.460329120747594, 2.166904282348720, 2.940398376294641, 3.552322552630517, 3.815113280296829, 4.718430329474392, 5.174732620979290, 5.352844880568423, 6.016840842917124, 6.568113524406891, 7.005129171962144, 7.571779540526922, 8.154051896209749, 8.309172276617504, 9.454823099621902, 9.603504233922508, 10.23394653129571, 10.57932305246922, 11.36710458175457, 11.60447466148275, 11.85943439844901, 12.46907894158106, 13.02956029114208, 13.23957982434998