L(s) = 1 | + 3-s − 3.29·5-s − 3.59·7-s + 9-s − 11-s + 13-s − 3.29·15-s − 8.12·17-s − 3.59·19-s − 3.59·21-s + 3.74·23-s + 5.83·25-s + 27-s + 3.69·29-s − 5.29·31-s − 33-s + 11.8·35-s − 9.82·37-s + 39-s + 8.08·41-s − 2.94·43-s − 3.29·45-s − 8.73·47-s + 5.90·49-s − 8.12·51-s − 8.58·53-s + 3.29·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.47·5-s − 1.35·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s − 0.849·15-s − 1.97·17-s − 0.824·19-s − 0.784·21-s + 0.781·23-s + 1.16·25-s + 0.192·27-s + 0.686·29-s − 0.950·31-s − 0.174·33-s + 1.99·35-s − 1.61·37-s + 0.160·39-s + 1.26·41-s − 0.448·43-s − 0.490·45-s − 1.27·47-s + 0.844·49-s − 1.13·51-s − 1.17·53-s + 0.443·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5434139545\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5434139545\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 + 3.59T + 7T^{2} \) |
| 17 | \( 1 + 8.12T + 17T^{2} \) |
| 19 | \( 1 + 3.59T + 19T^{2} \) |
| 23 | \( 1 - 3.74T + 23T^{2} \) |
| 29 | \( 1 - 3.69T + 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 + 9.82T + 37T^{2} \) |
| 41 | \( 1 - 8.08T + 41T^{2} \) |
| 43 | \( 1 + 2.94T + 43T^{2} \) |
| 47 | \( 1 + 8.73T + 47T^{2} \) |
| 53 | \( 1 + 8.58T + 53T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 61 | \( 1 - 7.24T + 61T^{2} \) |
| 67 | \( 1 - 4.38T + 67T^{2} \) |
| 71 | \( 1 + 9.33T + 71T^{2} \) |
| 73 | \( 1 + 2.99T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 4.09T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.044499349127200563832343424624, −7.14791885447494715854027841698, −6.80371916382468402327964353926, −6.09760941157462448614802996316, −4.80733807640413162702697218669, −4.26816339361242819182176339197, −3.46886979258412074696017677891, −3.01165092238520233557687253360, −1.97460818253951965966818265759, −0.34307159467423699625774946546,
0.34307159467423699625774946546, 1.97460818253951965966818265759, 3.01165092238520233557687253360, 3.46886979258412074696017677891, 4.26816339361242819182176339197, 4.80733807640413162702697218669, 6.09760941157462448614802996316, 6.80371916382468402327964353926, 7.14791885447494715854027841698, 8.044499349127200563832343424624