| L(s) = 1 | − i·3-s + i·5-s − 3.34·7-s − 9-s + 3.80·11-s − 4.01·13-s + 15-s + 3.43i·17-s + 2.65·19-s + 3.34i·21-s + (4.71 − 0.853i)23-s − 25-s + i·27-s + 2.73·29-s − 5.29i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447i·5-s − 1.26·7-s − 0.333·9-s + 1.14·11-s − 1.11·13-s + 0.258·15-s + 0.833i·17-s + 0.608·19-s + 0.730i·21-s + (0.984 − 0.178i)23-s − 0.200·25-s + 0.192i·27-s + 0.507·29-s − 0.950i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4592324757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4592324757\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-4.71 + 0.853i)T \) |
| good | 7 | \( 1 + 3.34T + 7T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 + 4.01T + 13T^{2} \) |
| 17 | \( 1 - 3.43iT - 17T^{2} \) |
| 19 | \( 1 - 2.65T + 19T^{2} \) |
| 29 | \( 1 - 2.73T + 29T^{2} \) |
| 31 | \( 1 + 5.29iT - 31T^{2} \) |
| 37 | \( 1 + 1.26iT - 37T^{2} \) |
| 41 | \( 1 + 0.928T + 41T^{2} \) |
| 43 | \( 1 + 6.49T + 43T^{2} \) |
| 47 | \( 1 - 1.22iT - 47T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 15.1iT - 59T^{2} \) |
| 61 | \( 1 + 5.10iT - 61T^{2} \) |
| 67 | \( 1 + 3.01T + 67T^{2} \) |
| 71 | \( 1 - 4.10iT - 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 0.329iT - 89T^{2} \) |
| 97 | \( 1 + 4.88iT - 97T^{2} \) |
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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
−8.435448017581276352171828870501, −7.41765908386225372173547682783, −6.99447015870422868504192267813, −6.37615173130722617665844495640, −5.82060534673802171623773279744, −4.77057542167481711102652334422, −3.78142603057964796551545598900, −3.10724943498579466747883897598, −2.31710376827217411002453449935, −1.15433813353815794570329005081,
0.13028509526854543567981328505, 1.36613098808502730586306638899, 2.83843474744155988386890731292, 3.26380391451142603506299138719, 4.25996774509241207086120636377, 4.95084945980816698648094481702, 5.60934119432680833652385943947, 6.67475436754522007603475777089, 6.93410666182511287178517721568, 7.911464441218220870916801370795
