L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 12-s − 13-s − 15-s + 16-s − 17-s + 18-s + 4·19-s + 20-s − 24-s + 25-s − 26-s − 27-s + 6·29-s − 30-s + 4·31-s + 32-s − 34-s + 36-s + 2·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 1.11·29-s − 0.182·30-s + 0.718·31-s + 0.176·32-s − 0.171·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.564513654\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.564513654\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.65012010489738, −11.92802635585934, −11.90735612547306, −11.44627412992213, −10.84143962407329, −10.33753109952268, −9.937147187387495, −9.725512960507893, −8.844281004404072, −8.544238632756206, −7.904004690567006, −7.318235706144831, −6.938872361028044, −6.424443363251332, −6.088336441243555, −5.410389773385048, −5.118083828976910, −4.685827440149215, −4.078033426527532, −3.544019357696652, −2.893600194417824, −2.448668396927230, −1.791210869007495, −1.116663934507898, −0.5522342857857815,
0.5522342857857815, 1.116663934507898, 1.791210869007495, 2.448668396927230, 2.893600194417824, 3.544019357696652, 4.078033426527532, 4.685827440149215, 5.118083828976910, 5.410389773385048, 6.088336441243555, 6.424443363251332, 6.938872361028044, 7.318235706144831, 7.904004690567006, 8.544238632756206, 8.844281004404072, 9.725512960507893, 9.937147187387495, 10.33753109952268, 10.84143962407329, 11.44627412992213, 11.90735612547306, 11.92802635585934, 12.65012010489738