| L(s) = 1 | + (1 + 1.41i)3-s − 1.41i·5-s + (2 − 1.73i)7-s + (−1.00 + 2.82i)9-s − 2.44·11-s + 2·13-s + (2.00 − 1.41i)15-s + 7.34·17-s + 4·19-s + (4.44 + 1.09i)21-s + 1.41i·23-s + 2.99·25-s + (−5.00 + 1.41i)27-s − 4.89·29-s − 6.92i·31-s + ⋯ |
| L(s) = 1 | + (0.577 + 0.816i)3-s − 0.632i·5-s + (0.755 − 0.654i)7-s + (−0.333 + 0.942i)9-s − 0.738·11-s + 0.554·13-s + (0.516 − 0.365i)15-s + 1.78·17-s + 0.917·19-s + (0.970 + 0.239i)21-s + 0.294i·23-s + 0.599·25-s + (−0.962 + 0.272i)27-s − 0.909·29-s − 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.97866 + 0.280862i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.97866 + 0.280862i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 7.34T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 4.89T + 29T^{2} \) |
| 31 | \( 1 + 6.92iT - 31T^{2} \) |
| 37 | \( 1 - 10.3iT - 37T^{2} \) |
| 41 | \( 1 + 2.44T + 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 5.65iT - 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 1.41iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 11.3iT - 83T^{2} \) |
| 89 | \( 1 + 7.34T + 89T^{2} \) |
| 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
−10.36718197437294745641456918849, −9.774731629754203357481961655843, −8.799298182005924127823869287651, −7.950107157575609838151928677787, −7.47934936396933530320695547223, −5.64401422171732604707484757459, −5.03148619022154688761349881395, −4.01421574802912427322002991972, −3.02463967675031995101286142866, −1.33854669547455588289807103362,
1.38821452724373150116078617447, 2.68322635374238142986754050833, 3.50151502224562105931430946350, 5.23612399819530042757456476739, 5.97019277122333039079928497463, 7.21860662694130983635897200666, 7.76439119491150104234192776074, 8.581134907387006318459571191486, 9.463473718342760177320295772379, 10.55638257930453475721858723496
