| L(s) = 1 | − 7-s + 3·11-s + 4·13-s − 3·17-s − 19-s − 6·29-s + 4·31-s − 2·37-s + 6·41-s − 43-s + 3·47-s − 6·49-s + 12·53-s − 6·59-s − 61-s − 4·67-s + 6·71-s + 7·73-s − 3·77-s − 8·79-s − 12·83-s − 12·89-s − 4·91-s − 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.904·11-s + 1.10·13-s − 0.727·17-s − 0.229·19-s − 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.937·41-s − 0.152·43-s + 0.437·47-s − 6/7·49-s + 1.64·53-s − 0.781·59-s − 0.128·61-s − 0.488·67-s + 0.712·71-s + 0.819·73-s − 0.341·77-s − 0.900·79-s − 1.31·83-s − 1.27·89-s − 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.33534483311632, −13.88024151650778, −13.45405178030935, −12.89092460324668, −12.53685613069622, −11.81931996061841, −11.36720079032667, −10.99314193361180, −10.41256499540213, −9.798349987295019, −9.279260408447154, −8.811151649177823, −8.419772383120887, −7.731514801444130, −7.031116053530077, −6.661734102589605, −6.021783827645021, −5.709744296395153, −4.833150766845366, −4.130657949426976, −3.836121774896243, −3.118459894491392, −2.393773551399162, −1.626577422742566, −0.9860139685960952, 0,
0.9860139685960952, 1.626577422742566, 2.393773551399162, 3.118459894491392, 3.836121774896243, 4.130657949426976, 4.833150766845366, 5.709744296395153, 6.021783827645021, 6.661734102589605, 7.031116053530077, 7.731514801444130, 8.419772383120887, 8.811151649177823, 9.279260408447154, 9.798349987295019, 10.41256499540213, 10.99314193361180, 11.36720079032667, 11.81931996061841, 12.53685613069622, 12.89092460324668, 13.45405178030935, 13.88024151650778, 14.33534483311632
