L(s) = 1 | − 127·3-s + 256·4-s − 625·5-s − 2.40e3·7-s + 9.56e3·9-s − 2.39e4·11-s − 3.25e4·12-s + 5.65e4·13-s + 7.93e4·15-s + 6.55e4·16-s + 9.78e4·17-s − 1.60e5·20-s + 3.04e5·21-s + 3.90e5·25-s − 3.81e5·27-s − 6.14e5·28-s − 8.51e4·29-s + 3.04e6·33-s + 1.50e6·35-s + 2.44e6·36-s − 7.18e6·39-s − 6.13e6·44-s − 5.98e6·45-s − 2.19e6·47-s − 8.32e6·48-s + 5.76e6·49-s − 1.24e7·51-s + ⋯ |
L(s) = 1 | − 1.56·3-s + 4-s − 5-s − 7-s + 1.45·9-s − 1.63·11-s − 1.56·12-s + 1.98·13-s + 1.56·15-s + 16-s + 1.17·17-s − 20-s + 1.56·21-s + 25-s − 0.718·27-s − 28-s − 0.120·29-s + 2.56·33-s + 35-s + 1.45·36-s − 3.10·39-s − 1.63·44-s − 1.45·45-s − 0.449·47-s − 1.56·48-s + 49-s − 1.83·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.8676892008\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8676892008\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + p^{4} T \) |
| 7 | \( 1 + p^{4} T \) |
good | 2 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 3 | \( 1 + 127 T + p^{8} T^{2} \) |
| 11 | \( 1 + 23953 T + p^{8} T^{2} \) |
| 13 | \( 1 - 56593 T + p^{8} T^{2} \) |
| 17 | \( 1 - 97873 T + p^{8} T^{2} \) |
| 19 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 23 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 29 | \( 1 + 85153 T + p^{8} T^{2} \) |
| 31 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 37 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 47 | \( 1 + 2191487 T + p^{8} T^{2} \) |
| 53 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 59 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 61 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 67 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 71 | \( 1 - 50742722 T + p^{8} T^{2} \) |
| 73 | \( 1 + 33491522 T + p^{8} T^{2} \) |
| 79 | \( 1 - 70135727 T + p^{8} T^{2} \) |
| 83 | \( 1 - 54186718 T + p^{8} T^{2} \) |
| 89 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 97 | \( 1 - 165613873 T + p^{8} T^{2} \) |
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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
−15.49470455365525980026450965828, −13.03241765313294158017206966637, −12.10612618452883997387408457053, −11.05057766900425893256253773156, −10.41019801330753700972076359557, −7.88517604940027651696196071776, −6.54941588225473332706940654949, −5.53036583221216933997998678278, −3.38865216048735633692219070862, −0.73710297682374860170140703100,
0.73710297682374860170140703100, 3.38865216048735633692219070862, 5.53036583221216933997998678278, 6.54941588225473332706940654949, 7.88517604940027651696196071776, 10.41019801330753700972076359557, 11.05057766900425893256253773156, 12.10612618452883997387408457053, 13.03241765313294158017206966637, 15.49470455365525980026450965828