| L(s) = 1 | − 7-s − 5·11-s − 4·13-s − 17-s − 19-s − 9·29-s + 6·31-s + 3·37-s − 5·41-s + 2·43-s − 9·47-s − 6·49-s + 3·53-s − 6·59-s − 14·67-s − 8·71-s − 7·73-s + 5·77-s + 6·79-s − 12·83-s + 6·89-s + 4·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.50·11-s − 1.10·13-s − 0.242·17-s − 0.229·19-s − 1.67·29-s + 1.07·31-s + 0.493·37-s − 0.780·41-s + 0.304·43-s − 1.31·47-s − 6/7·49-s + 0.412·53-s − 0.781·59-s − 1.71·67-s − 0.949·71-s − 0.819·73-s + 0.569·77-s + 0.675·79-s − 1.31·83-s + 0.635·89-s + 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.00211692955060, −14.37732812034928, −13.60287903692393, −13.16937930664598, −12.98025623019244, −12.27855214637407, −11.82292914070544, −11.22467185815295, −10.66558314573561, −10.17973424957224, −9.752921321642676, −9.266125591431149, −8.552643952931804, −8.023025985110107, −7.480956996587155, −7.155752825724766, −6.311455880425027, −5.889029710282973, −5.115448369391534, −4.813611694695850, −4.121279820701207, −3.253137047198229, −2.769034906388990, −2.205631463389368, −1.408162103618530, 0, 0,
1.408162103618530, 2.205631463389368, 2.769034906388990, 3.253137047198229, 4.121279820701207, 4.813611694695850, 5.115448369391534, 5.889029710282973, 6.311455880425027, 7.155752825724766, 7.480956996587155, 8.023025985110107, 8.552643952931804, 9.266125591431149, 9.752921321642676, 10.17973424957224, 10.66558314573561, 11.22467185815295, 11.82292914070544, 12.27855214637407, 12.98025623019244, 13.16937930664598, 13.60287903692393, 14.37732812034928, 15.00211692955060
