| L(s) = 1 | − 1.73i·3-s + 4·7-s − 2.99·9-s − 6.92i·13-s + (−4 + 1.73i)19-s − 6.92i·21-s + 5·25-s + 5.19i·27-s + 10.3i·31-s + 6.92i·37-s − 11.9·39-s + 8·43-s + 9·49-s + (2.99 + 6.92i)57-s − 14·61-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s + 1.51·7-s − 0.999·9-s − 1.92i·13-s + (−0.917 + 0.397i)19-s − 1.51i·21-s + 25-s + 0.999i·27-s + 1.86i·31-s + 1.13i·37-s − 1.92·39-s + 1.21·43-s + 1.28·49-s + (0.397 + 0.917i)57-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10822 - 0.727782i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10822 - 0.727782i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 19 | \( 1 + (4 - 1.73i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 6.92iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 17.3iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−12.24304710224469179937119681422, −11.06586105635352166972471701874, −10.46569545040292379757921209993, −8.605968145022203165711821729466, −8.135402967216230923227603521132, −7.17703168599420294922936329146, −5.83614202168839100570342329163, −4.86026917588330792405449170020, −2.90592268459195561072923380780, −1.32745434870578809050888299030,
2.16990204724565111369613215422, 4.18015746641487345064874672619, 4.72859829669911814153244487782, 6.10368085663466887739813171474, 7.54718937664246696155688095968, 8.742067627107389086516496909394, 9.312495513835628673184773541847, 10.72782135161482387801569156565, 11.24026530167860742716111213154, 12.05146778393458729062899157995
