L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s + 11-s − 12-s + 3·13-s − 14-s + 16-s − 4·17-s − 18-s − 21-s − 22-s − 7·23-s + 24-s − 3·26-s − 27-s + 28-s + 3·31-s − 32-s − 33-s + 4·34-s + 36-s − 8·37-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 + 0.301·11-s − 0.288·12-s + 0.832·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.218·21-s − 0.213·22-s − 1.45·23-s + 0.204·24-s − 0.588·26-s − 0.192·27-s + 0.188·28-s + 0.538·31-s − 0.176·32-s − 0.174·33-s + 0.685·34-s + 1/6·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−12.52125025239579, −12.14004199097595, −11.61174420663800, −11.39731425977842, −10.88268492406506, −10.46787067322770, −10.04201221180429, −9.608209481017620, −9.040604604185608, −8.545306440643600, −8.249511550170196, −7.809485379118088, −7.061239251226268, −6.781738083204466, −6.311973880451081, −5.844646673037693, −5.347342455516865, −4.774083510759207, −4.175014356862802, −3.752288063754461, −3.191695194292512, −2.220379342146202, −2.022471129601523, −1.304911406042071, −0.6979674552618533, 0,
0.6979674552618533, 1.304911406042071, 2.022471129601523, 2.220379342146202, 3.191695194292512, 3.752288063754461, 4.175014356862802, 4.774083510759207, 5.347342455516865, 5.844646673037693, 6.311973880451081, 6.781738083204466, 7.061239251226268, 7.809485379118088, 8.249511550170196, 8.545306440643600, 9.040604604185608, 9.608209481017620, 10.04201221180429, 10.46787067322770, 10.88268492406506, 11.39731425977842, 11.61174420663800, 12.14004199097595, 12.52125025239579