| L(s) = 1 | − 2.46·5-s + 5.48i·7-s − 10.5i·11-s − 8.96·13-s + 16.2·17-s − 2.96i·19-s + 1.46i·23-s − 18.9·25-s + 25.0·29-s − 10.5i·31-s − 13.5i·35-s + 16.9·37-s + 29.0·41-s − 34.9i·43-s + 86.1i·47-s + ⋯ |
| L(s) = 1 | − 0.492·5-s + 0.783i·7-s − 0.962i·11-s − 0.689·13-s + 0.955·17-s − 0.156i·19-s + 0.0635i·23-s − 0.757·25-s + 0.865·29-s − 0.339i·31-s − 0.385i·35-s + 0.458·37-s + 0.707·41-s − 0.813i·43-s + 1.83i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.598249993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.598249993\) |
| \(L(2)\) |
|
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 | \( 1 + 2.46T + 25T^{2} \) |
| 7 | \( 1 - 5.48iT - 49T^{2} \) |
| 11 | \( 1 + 10.5iT - 121T^{2} \) |
| 13 | \( 1 + 8.96T + 169T^{2} \) |
| 17 | \( 1 - 16.2T + 289T^{2} \) |
| 19 | \( 1 + 2.96iT - 361T^{2} \) |
| 23 | \( 1 - 1.46iT - 529T^{2} \) |
| 29 | \( 1 - 25.0T + 841T^{2} \) |
| 31 | \( 1 + 10.5iT - 961T^{2} \) |
| 37 | \( 1 - 16.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 29.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 34.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 86.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 80.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 66.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 0.966T + 3.72e3T^{2} \) |
| 67 | \( 1 - 113. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 51.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 79.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 142.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 45.8T + 9.40e3T^{2} \) |
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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
−9.563899351103301592783720309636, −8.730869565239081058498385910288, −8.002341876460287475560571533436, −7.26510620937425800513493565890, −6.06924173908629858893291313108, −5.48404082278796482005556864833, −4.36838360799873745190653255254, −3.30232723399910373336582398450, −2.37090685504707112094839066227, −0.73728069931464738427823381674,
0.78138809249653784476987384669, 2.21176566364685470740950149031, 3.51190127041677113456299629614, 4.35381944674247013120664382885, 5.18885636999530854626515115711, 6.36934254356061701808212323799, 7.39563189044469337101404107025, 7.66899598684893199602299766619, 8.753254823976688078144907173313, 9.897723364505938004837586189928
