| L(s) = 1 | + 2i·5-s + 6.92i·7-s + 6.92·11-s − 2i·13-s − 10·17-s + 20.7·19-s − 27.7i·23-s + 21·25-s − 26i·29-s + 6.92i·31-s − 13.8·35-s + 26i·37-s + 58·41-s + 48.4·43-s − 69.2i·47-s + ⋯ |
| L(s) = 1 | + 0.400i·5-s + 0.989i·7-s + 0.629·11-s − 0.153i·13-s − 0.588·17-s + 1.09·19-s − 1.20i·23-s + 0.839·25-s − 0.896i·29-s + 0.223i·31-s − 0.395·35-s + 0.702i·37-s + 1.41·41-s + 1.12·43-s − 1.47i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.223576312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.223576312\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2iT - 25T^{2} \) |
| 7 | \( 1 - 6.92iT - 49T^{2} \) |
| 11 | \( 1 - 6.92T + 121T^{2} \) |
| 13 | \( 1 + 2iT - 169T^{2} \) |
| 17 | \( 1 + 10T + 289T^{2} \) |
| 19 | \( 1 - 20.7T + 361T^{2} \) |
| 23 | \( 1 + 27.7iT - 529T^{2} \) |
| 29 | \( 1 + 26iT - 841T^{2} \) |
| 31 | \( 1 - 6.92iT - 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 58T + 1.68e3T^{2} \) |
| 43 | \( 1 - 48.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 69.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 74iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 90.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + 26iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 6.92T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 46T + 5.32e3T^{2} \) |
| 79 | \( 1 - 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 48.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 82T + 7.92e3T^{2} \) |
| 97 | \( 1 - 2T + 9.40e3T^{2} \) |
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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
−9.013427668841625637550106716452, −8.230101673181972429797429912992, −7.34608833044142979241859484668, −6.51063582142869721302433146109, −5.90395211089713032119940164207, −4.97424748181450191290868539251, −4.08486040669370148244136438803, −2.94807765796001552132122540498, −2.29457791482618979114251267847, −0.894291396568420371438610688122,
0.73248850573389047262912518206, 1.57628357372537341357225596994, 3.01108482372970073198034129232, 3.92708603298862517069102243335, 4.62990701588148983351473568654, 5.54890409103611305397396946473, 6.46207339599486713950923693547, 7.34793030861823519584973759845, 7.72340527551436006193855578515, 9.026190828816747002452542126225
