L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 3·11-s − 12-s − 4·13-s − 14-s + 16-s − 2·17-s − 18-s − 21-s + 3·22-s + 4·23-s + 24-s + 4·26-s − 27-s + 28-s + 3·31-s − 32-s + 3·33-s + 2·34-s + 36-s + 7·37-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 − 0.904·11-s − 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.218·21-s + 0.639·22-s + 0.834·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.188·28-s + 0.538·31-s − 0.176·32-s + 0.522·33-s + 0.342·34-s + 1/6·36-s + 1.15·37-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 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.64127966957393, −12.07738429327154, −11.70802666978212, −11.36084459413153, −10.84895743294960, −10.42910707410153, −9.986997232626984, −9.703353741899633, −9.124551481139241, −8.469490903765272, −8.257074018795316, −7.649568593982833, −7.236837980774485, −6.759366119740388, −6.446035275679322, −5.621050879525018, −5.297262584194418, −4.826093753087604, −4.417577499329293, −3.641290637452613, −2.967283698957433, −2.465037058891381, −2.005076041198658, −1.285614352640145, −0.6062791412328350, 0,
0.6062791412328350, 1.285614352640145, 2.005076041198658, 2.465037058891381, 2.967283698957433, 3.641290637452613, 4.417577499329293, 4.826093753087604, 5.297262584194418, 5.621050879525018, 6.446035275679322, 6.759366119740388, 7.236837980774485, 7.649568593982833, 8.257074018795316, 8.469490903765272, 9.124551481139241, 9.703353741899633, 9.986997232626984, 10.42910707410153, 10.84895743294960, 11.36084459413153, 11.70802666978212, 12.07738429327154, 12.64127966957393