| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 11-s + 2·13-s − 14-s + 16-s − 5.12·17-s − 7.12·19-s − 20-s − 22-s − 1.12·23-s + 25-s + 2·26-s − 28-s + 8.24·29-s + 1.12·31-s + 32-s − 5.12·34-s + 35-s − 1.12·37-s − 7.12·38-s − 40-s + 0.876·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.554·13-s − 0.267·14-s + 0.250·16-s − 1.24·17-s − 1.63·19-s − 0.223·20-s − 0.213·22-s − 0.234·23-s + 0.200·25-s + 0.392·26-s − 0.188·28-s + 1.53·29-s + 0.201·31-s + 0.176·32-s − 0.878·34-s + 0.169·35-s − 0.184·37-s − 1.15·38-s − 0.158·40-s + 0.136·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.397409394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.397409394\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 - 0.876T + 41T^{2} \) |
| 43 | \( 1 - 3.12T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 + 0.876T + 67T^{2} \) |
| 71 | \( 1 - 7.12T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 - 2.87T + 83T^{2} \) |
| 89 | \( 1 + 8.24T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.956623247277752467240851884730, −7.01478624352382966444527285185, −6.51555258531452685652515508880, −5.94331362830668814636361955724, −4.99179382775947233077391195472, −4.23422571321895000833919778921, −3.83097349435445121936693327273, −2.71200052628488644542851369707, −2.14583186348949781335329308231, −0.68526372017981084398745805941,
0.68526372017981084398745805941, 2.14583186348949781335329308231, 2.71200052628488644542851369707, 3.83097349435445121936693327273, 4.23422571321895000833919778921, 4.99179382775947233077391195472, 5.94331362830668814636361955724, 6.51555258531452685652515508880, 7.01478624352382966444527285185, 7.956623247277752467240851884730
