| L(s) = 1 | + 7-s − 5·11-s − 13-s − 5·17-s + 7·29-s + 9·31-s − 8·37-s + 2·41-s − 8·43-s − 9·47-s − 6·49-s − 11·53-s + 59-s − 7·61-s + 15·67-s − 8·71-s + 4·73-s − 5·77-s + 4·79-s − 9·83-s − 16·89-s − 91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.50·11-s − 0.277·13-s − 1.21·17-s + 1.29·29-s + 1.61·31-s − 1.31·37-s + 0.312·41-s − 1.21·43-s − 1.31·47-s − 6/7·49-s − 1.51·53-s + 0.130·59-s − 0.896·61-s + 1.83·67-s − 0.949·71-s + 0.468·73-s − 0.569·77-s + 0.450·79-s − 0.987·83-s − 1.69·89-s − 0.104·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.013178422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013178422\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.53922737976128, −14.00672302674831, −13.63019245676803, −13.00354079767420, −12.67775592229813, −11.99960200701639, −11.45626606789538, −10.98423787626966, −10.40866891938719, −10.00514710216726, −9.471910574532064, −8.610938203325081, −8.189011241200523, −7.973274688615076, −7.013727857193436, −6.666254717659236, −6.027233105378678, −5.195573364409513, −4.770647509746203, −4.454116471591201, −3.325305461406003, −2.855459461487458, −2.183846821924556, −1.475009960691060, −0.3465148229577240,
0.3465148229577240, 1.475009960691060, 2.183846821924556, 2.855459461487458, 3.325305461406003, 4.454116471591201, 4.770647509746203, 5.195573364409513, 6.027233105378678, 6.666254717659236, 7.013727857193436, 7.973274688615076, 8.189011241200523, 8.610938203325081, 9.471910574532064, 10.00514710216726, 10.40866891938719, 10.98423787626966, 11.45626606789538, 11.99960200701639, 12.67775592229813, 13.00354079767420, 13.63019245676803, 14.00672302674831, 14.53922737976128
